Learn German: Dieser graue Vogel kann nicht das Männchen sein.

This is the subjective use of können. Instead of talking about someone's physical ability to do something, it expresses a logical deduction. When combined with nicht, it means you are certain something is impossible, just like saying "that can't be true" in English.

Could we use muss nicht here instead of kann nicht to mean "must not be"?

No, and this is a very common trap! In German, muss nicht means "doesn't have to" (a lack of obligation). To express the strong deduction that something is impossible (what English speakers often call "must not be" or "cannot be"), you always use kann nicht.

Why is the adjective ending graue instead of grauer for a masculine noun?

Vogel is a masculine noun in the nominative case here (it is the subject). Because the preceding word dieser already shows the strong masculine ending (-er), the adjective only takes the "weak" ending, which is just -e.

Why does das Männchen use the neuter article das if it literally means "the male"?

In German, grammar rules often override biological gender. Any noun that ends in the diminutive suffix -chen is automatically grammatically neuter. This is the exact same rule that makes das Mädchen (the girl) neuter.