Gender is grammatical, so all words have genders, even words for inanimate objects. It's fairly predictable, though, based off of the end letter, as the other commentator said.

More generally, the possessive pronouns in Ukrainian ("my", "your", "their", etc.) function grammatically as adjectives. Each of them has 24 different forms for each combination of gender (masculine, neuter, feminine, and plural) and case (nominative, genitive, dative, accusative, instrumental, locative), and must match the gender & case of the word being modified.

So for example -

This is my house. Це (this) мій (my - masculine & nominative) дім (nominative).

vs

I go to my house. Я (I) йду (go - first person singular) до (towards) моєму (my - masculine & dative) дому (dative).