I've been really wondering about this, because I've noticed it several times with this and a few other words (I can't think of them right now). By "adjective-like words", I assume you mean that possessive pronouns follow the rules for declining masculine nouns, even though you change the endings as if the noun itself were feminine.
For example, папа in Dative case would be папе which is the feminine singular Dative ending for a word ending in a in Nominative case. The Dative case declension for "my" is:
So, under the grammar rules, "my dad" in Dative case would be моему папе and not моей папе, correct?
BUT apart from the formal use directed towards one person, you also use "вы/ваш" when you're talking about or to a group of people. Say, you want to express that a group of people, e.g. sisters have one dad, you say (directly to the group) "это ваш папа", which in english is also translated with "this is your Dad"
I've been drilling these lessons - so repeating them a lot. Every single time I answer 'mum and dad' and get dinged for it! It seems to me the way translations should really work is: 1) read sentence(s); 2) derive meaning; 3) construct natural equivalent in the target language.... 'dad & mum' just hurts! But I suppose they're also verifying we know how the nouns map, so therein lies the quandary.