Can FeedAPI store RSS feeds for individual users?

You can do that, but you need additional module.
I think with views you can solve the problem. The only thing is that you need to write an additional handler.
It's not trivial, you need to store per-user feed selection.
I do not plan to include such feature into main FeedAPI package.
Btw. if someone knows better way to do this, let's share :) I'm not really familiar with views.

I'm trying to do this very thing.

I'm using Fivestar/VotingAPI as a "one star" widget that allows users to "subscribe" to a feed by voting, and "unsubscribe" by cancelling their vote. This way I can get a view of the feeds a user has subscribed to by using the "(Individual votes) Voting API votes: Current user" filter. Can't select the user by argument (in VotingAPI 6.x-2.0-rc1), which is irritating.

What I would like to do is get a date-sorted teaser list of the feed Items (I'm using FeedAPI Node to store items) from all the feeds a used has subscribed to. This doesn't appear to be possible out of the box. The feed items don't inherit a vote from their parent feed, and I wouldn't want them to because I want to use votes for a different purpose on the items. I think the solution will involve using relationships in some way, which is bad news for me because complex database-y stuff makes my head hurt.

The current version will not because it has no relation between feed and feed item. If you grab the patch over on #347526: Full views support, admin/content/feed powered by views and apply it you'll be able to create a relation from feed items to their feeds. You'll want the relationship to be marked as required. Then you'd add a argument for User Id and tell it to use the Parent feed relationship. This way when you provide a uid it's matching the Feed's uid and the nodes you see will be the feed items associated with the feeds the user created.