Is this even possible?

Of course it's possible. It sounds fun! Some years ago I was working on a series of children's short stories about a guy and his dog, from the dog's POV. It was some of the most fun I've had writing.

When I asked essentially the same question, the input I got from here was to publish separately as many of the shorts as possible in magazines first, then compile them later into books. I was told this was much easier to do than to try to sell a collection of unpublished short stories, which few publishers will do. Just make sure what right the short story market takes, like only taking first North American rights or that have the rights revert to you after a year or so. Otherwise you can't republish in a book later.

Others have done it, so why not? There is Conan from Robert E. Howard for example. Guin Saga is also a series of novels with the same characters, there is actually more than a hundred books. So, of course it is possible.

The best way to do this is to sell the best stories one at a time to the appropriate genre magazines or anthologies. You might find that one place might be happy to take a second piece if you have a first published with them.

A lot of authors will re-use characters in their short stories, so it's quite common.

Once you've got a bit of a list of sales up (4 or 5 published), you could shop for a publisher for a book-sized collection (published and unpublished stories). Some digital-first publishers have been experimenting with serial fiction for several years now.

Depending on which audience you are shooting for, it should definitely work. The best stories will grab a wide variety of attention but the lovers will come back again and again to watch the characters develop in different circumstances. I know myself as a reader and I get attached to a style and cant shake it so it really should work!