Learn Spanish (Latin America): El gemelo menor es igual de tímido que su hermana. - The younger twin is as shy as his sister.

Yes! Both mean "as shy as" and they are completely interchangeable when you are comparing qualities like adjectives or adverbs. Just be careful not to mix up the pieces: it is always igual de... que and tan... como.

Since we are comparing him to his sister, why is the adjective tímido and not tímida?

The adjective always has to agree with the main subject of the sentence, which is the person or thing coming before the verb. Here, the subject is El gemelo menor (masculine), so we use the masculine adjective tímido. The sister (su hermana) is just the point of reference at the end.

Why is it menor instead of más joven for "younger"?

While más joven is perfectly understood, menor is the standard, more natural way to say "younger" (and mayor for "older") when talking about siblings or age in general. Notice that like most describing words, it goes after the noun (gemelo menor).

If the sentence was about the younger female twin, would igual de change to match her gender?