I noticed that when purchasing the bundle, it doesn’t recognize the assets I already own, so it seems that I might be paying for those assets again.
Is there a way to solve this problem?

Now that would be something. But you should check if your bundle even gives you any discount. Bundles on Itch are just that. Bundles. There is no need for a discount to bundle things together.

This is hard to recognise, if the individual items are on discount, because they are on halloween sale for example. Or when there are free items in a bundle.

While they are at it, they might want to implement a shopping cart that would at least bundle the things together that can be paid together (which is an advantage to bundle things without discount in the first place) and let you increase the donation level of a game for individual priced files, instead of making you buy the same game twice. I know how this is solved on Steam. You simply cannot buy the same thing twice. You could only gift it, for the second time. And if you buy a bundle with something you already have, the discounted bundle price is adjusted. But as far as I know every bundle on Steam always has a discount.