It really has to be
If he is 21 this year, he must have been 20 last year.
because the past tense applies logically to "he be 20 last year". Using an adverb to express the "must" part, we'd have
Necessarily, if he is 21 this year, he was 20 last year.
where the past tense of the clause "he be 20 last year" is expressed by using the past tense form of "be", which is normal in English.
When instead of "necessarily", we use "must", you might expect the second clause to be expressed
*He must was 20 last year.
But here we run into some idiosyncrasies of English grammar. You can't have a tensed verb like "was" following a modal verb; English permits only tenseless, i.e. non-finite, verb forms after a modal. And in a position where a tense inflection is not allowed, a present tense is just lost, but a past tense is converted to perfect "have". That's why we get "He must have been 20 last year". This is not logically a perfect; it's a substitute for a past tense which otherwise could not be expressed.
You can also see this conversion of a past tense to a perfect in some infinitive verb complements. "Believe" takes either a "that"-clause complement or an infinitive complement:
I believe that Mars is red.
I believe Mars to be red.
Notice that the present tense of "is" is simply not expressed in the infinitive form. But in a past tense complement,
I believe that Mars was watery at one time.
I believe Mars to have been watery at one time.
the logical past tense turns up as a perfect, because a tense is not permitted in a "to"-infinitive in English.