Would one turn into the quality downvote while the other turned into the feelings downvote?

No, they wouldn't. The problem is that if considered intrapersonal, different buttons could get different qualities, but when considered interpersonal they probably wouldn't.

There are different possible ways to fill multiple downvote buttons with a sense. For example, it could be "feelings downvote" and "quality downvote". Or "quality downvote" and "feelings downvote" from left to right. Or it could be "slight downvote" and "strong downvote" and "slight+strong downvote" which is even stronger. Different people will use buttons differently, so in average it would be the same mess. Probably if a PhD dug into the raw data, she could make sense from it by categorizing downvoters and summing up their votes with normalizing coefficients, but I'm not sure.

Lichess recently changed their puzzles and forced me to vote for each puzzle. I really do not know which puzzle is good and which is not. I struggled to find a way to vote, and finally came to a simple strategy "upvote each puzzle", because I do not know any better. I'm sorry I did it, I wish to help lichess, but do not know how to. When faced with two downvote buttons I probably would struggle to make sense of it, then I'd find some simple strategy which I could verbalize and stick to it. It would be improbable, that the strategy would be "feelings" vs "quality", because how could I separate my feelings from a perceived quality?

It just doesn't matter that much. No one (well, save literally a single person) can even see vote totals. Votes do one thing—make comments more or less prominent within a thread.

I see voting as a sort of collective curation mechanism, and I vote accordingly. If I see a insightful comment that I think more people should read, I upvote it. If I see a dumb comment I think less people should read, I downvote it. Assuming others agree with me, the insightful comments end up higher in the thread, and the stupid ones become faded and easy to ignore.