Fun fact: CollectaCards give you bonuses. The game tells you this information, but doesn't go into details. Luckily, Reddit plays this game too much!
Summon cards give a bonus to that summon's damage, with a cap at 1.3 increase. Enemy cards give a bonus to that enemy's experience reward, with a cap at 1.2 increase. Character cards give a bonus to every stat except HP, with no cap besides the general cap of 999. One person on Reddit confirmed this by
getting a couple of characters to 999s at level 1. You wouldn't want to do that, of course, because it takes a lot longer than grinding someone to 9 stars, and 9 stars will get their stats to like 600-700 anyway.
Also, I'm going to spoiler this next bit because maybe people want to figure this mechanic out on their own, but
drops aren't completely random. Songs in a particular series are more likely to drop cards from that series. Which is great news, because it means you have a modicum of control over cards you're trying to target. I think this applies across sequels, so all the FF VII games bias drops for every FF VII game, but I'm not 100% sure about that.