Learn Portuguese (Portugal): O pássaro é tão pequeno como o rato.

You use the structure tão + [adjective] + como. Here, tão means "as" (or "so") and como means "as" (or "like"). Put them together around the adjective pequeno, and you get tão pequeno como (as small as).

Does the word pequeno need to match both animals?

No, it only needs to match the first noun, which is the subject of the sentence. Since o pássaro is masculine and singular, we use pequeno. If the first animal were feminine, like a spider (a aranha), the adjective would change to pequena, even if the second animal is still a masculine mouse.

I remember como means "how". Why is it used here?

The word como is very flexible and has a few different meanings depending on the context. It means "how" when asking a question, but it also means "like" or "as" when making comparisons like this one.

Why use é from the verb ser rather than a form of estar?

We use ser for permanent or inherent characteristics. Since being small is a natural, defining trait of this bird rather than a temporary state, é is the correct verb to use.