tl;dr – Short answer

Your example is grammatical because the reflexive pronoun has an antecedent (a noun phrase with which it is coreferential) and both are complements of the same verb. The antecedent is the indirect object (IO) and the reflexive is the direct object (DO), and in this construction reflexives are mandatory.

Long answer

Types of reflexive pronouns

Reflexive pronouns have various different functions, depending on sentence structure and what the pronoun is governed by. If you have access to it, there is a very thorough description of them in The Cambridge Grammar of the English Language by Rodney Huddleston and Geoffrey K. Pullum (pp. 1483–1499); the following is mostly excerpted from their description.

What CGEL calls basic reflexives can only be used if there is a close, structural link between the reflexive pronoun and its antecedent. The antecedent can be any kind of noun phrase (including a pronoun), and it can be covert as well than overt (i.e., it doesn’t have to actually appear as a word in the sentence) – but there must be a link between the two.¹

Reflexive constructions

The two most common constructions involving basic reflexive pronouns are those where the pronoun and the antecedent are linked –

1. directly through a shared relationship to the same verb (both are complements of the verb), or


2. more indirectly through sentence constituents that are somehow related to the same verb.

2 includes a host of syntactically more complex cases, of which the most common is where the reflexive is the object of a preposition and the prepositional phrase as a whole is a complement of the verb (I was running from myself). But we don’t need to deal with 2 here, because your example falls under 1.

1 includes all those cases where both entities (pronoun and antecedent) are in themselves complements of the same verb. That verb can be the main verb in a sentence, or it can be in a subordinate clause. The archetypal version is where the antecedent is the subject and the reflexive is the direct object – that’s the one you’ve learnt from your teachers, and the one most online sources will give. But it’s not the whole truth; in fact, it’s only a small part of the whole truth. The fact of the matter is that it doesn’t really matter what the syntactic function of the two entities is. Here are some examples of the possible constructions where both antecedent and pronoun are complements of the verb (shown as antecedent + pronoun):

• 
  S(ubject) + D(irect) O(bject): I injured myself, Sue injured herself
  


• 
  S + I(ndirect) O: I bought myself a book, John bought himself a book
  


• 
  S + P(redicative) C(omplement): I am myself, Ann is not feeling herself today
  


• 
  DO + PC: I call him himself²
  


• 
  IO + DO: Art shows us ourselves
  

As you may have noticed, the last item on that list (IO + DO) fits your example to a T.

In I showed the monkey himself in a mirror, the antecedent is the monkey, the indirect object, and the pronoun is himself, the direct object. Both are complements of the verb show, and it is this close, structural link between the two constituents which requires us to use the reflexive pronoun.

Possible non-reflexives in the same constructions

In many cases, a non-reflexive pronouns can also be used instead of a reflexive pronoun. There are two possible reasons for this: either the reflexive is optional (i.e., you can switch between the two with no difference in meaning in a given construction), or it can be overridden by a non-reflexive pronoun; in both cases, there is a tendency that non-reflexive forms are more common in the first and second persons and rarer in the third person.

As an example of the former, there are some dialects, especially in the US, where S + IO tends to be only optionally reflexive, so ^!I bought me a car and I bought myself a car are both possible and roughly equivalent. In the rest of the Anglosphere, the former of these is not possible (the raised exclamation mark means ‘non-standard’). In the IO + DO construction like your example, however, reflexives are mandatory.

Non-reflexive pronouns which ‘take over’ from reflexive ones are considered by CGEL to be part of the same group of override reflexives mentioned in note 1. They are called override reflexives because they can be used even where the use of reflexives is mandatory – they quite literally force-override the required form. They occur mostly in informal speech, but are not limited to specific dialects. These non-reflexives add contrast and emphasis: I injured myself is a neutral statement that I did something that resulted in an injury to my own body, whereas I injured me emphasises the contrast between my injuring myself and my injuring someone else.

Notes:

^{1 This is in opposition to override reflexives, which do not require any such link and can appear with no antecedent at all. Your example here contains a basic reflexive, so we can skip the override reflexives. An example of an override reflexive would be “Sarah drove John and myself to the park”, where myself has no antecedent at all and could just as well have been the regular personal pronoun me.}

^{2 I can’t think of any reason why this construction shouldn’t be grammatical, but I will note that it is very, very infrequent. I cannot think of an example that sounds natural and likely to ever be uttered in actual speech, but I believe this is because it is exceedingly rare that we ever need to express anything where a direct object has a predicative complement which is a pronoun.}

Within Generative Grammar frameworks, e.g. Government and Binding (Chomsky 1981), the answer to your question would run along the following lines:

The word "himself" is a reflexive anaphor. Such anaphors distribute according to Binding Principle A. Binding Principle A states:

An anaphor must be bound by an antecedent locally within its binding domain. Being bound means c-commanded, sharing in appropriate features, and co-indexed. A binding domain in English is, roughly, a clause with a subject position.

In your example, the conditions given in Binding Principle A are met. Therefore, the sentence is explained as grammatical.
_{(However, I find "show X Y" much worse than "show X to Y". "I showed Bill Mary" just like "I showed Bill himself" is worse for me than "I showed Bill to Mary" or "I showed Bill to himself". I'm therefore going to add "to" here)}

• The anaphor "himself" is bound by "the monkey" because it is c-commanded by it, it shares the features 3rd person singular, masculine, and can be co-indexed, shown by subscript i.

(1) [I showed [ [the monkey]_i [to himself_i ]]]
("the monkey"=antecedent, which correctly binds "himself"=anaphor)

Note, if any of the conditions are violated the sentence is correctly predicted to become ungrammatical. So, if the anaphor is not c-commanded by the antecedent, the sentence becomes impossible.

(2) *[[I showed [the monkey]_i around] [by swinging himself_i over my head]]
("the monkey" does not c-command "himself")

If the anaphor does not share the right features with the antecedent, the sentence also becomes impossible.

(3) *[I showed [the monkey]_i to themselves_i]
("the monkey" is 3rd person singular, but "themselves" is 3rd person plural)

• The anaphor "himself" is bound within its local binding domain. This domain, here, is simply the entire clause because that's where the subject "I" occurs (simplified).

(4) [_{binding domain}I showed [the monkey]_i to himself_i ]
("himself" is bound locally, WITHIN its binding domain)

Again, if this requirement is not met, the sentence becomes ungrammatical. For example, in the sentence below, the anaphor is not bound within its local binding domain - the antecendet is "too far" away - and so the sentence is impossible.

(5) *[I showed [the monkey]_i that [_{binding domain} bananas are good for himself_i ]]
("himself" is not bound locally, OUTSIDE of its binding domain)

(Examples of this kind, ditransitives with indirect objects binding a direct object anaphor, are important because they are used as evidence for certain structural assumptions, i.e. for an aysmmetric structure of objects within the VP, for verb movement, for VP shells, or for little vP.)

We could say that "Art shows us ourselves in its mirror". I think that's grammatical, and by analogy, "I showed the monkey itself in the mirror" would be grammatical, but I think many native speakers would avoid it and say "its image" instead of "itself". Maybe that's because of the monkey. This seems fine to me:

The town's most popular dressmaker was accused of showing customers themselves in mirrors that flatter.

though I suspect a good number of speakers would choose to say it this way:

The town's most popular dressmaker was accused of showing customers to themselves in mirrors that flatter.

This is not an answer about why it is grammatical, but to show the actual usage of the phrase. Note that the monkey is referred to by "him", not "it". My reading of the sentence is the speaker talking about a man whom (s)he is likening to a monkey. The speaker showed this man his true nature either through an act which made him reflect, or by reflecting his behavior him/herself to show him "look, this is how you behave". The mirror is a metaphor. The original sentence, in my opinion, can be rephrased as "oh I showed him what a monkey he really is".

So yes, not only does the sentence grammatically sound as others have explained wonderfully, but the usage also makes perfect sense to me in the right context.

"[to] the monkey" is an indirect object;
"himself in the mirror" is the direct object.
("to" is used only to clarify the indirect nature.)
One could say "... show [to] himself the monkey in the mirror"; but (1) it is not idiomatic; and because of the phrase "in the mirror", it becomes very misleading, even ambiguous. Besides, putting "the monkey in the mirror" before "to the monkey" merely exacerbates that problem.
As soon as you see what the parts are doing, you see that it is grammatical.
One cannot really say "show the monkey the monkey in the mirror" for this usage; it would suggest that the mirror is showing a different monkey.