Best I can do to help with the "feel" thing is to pull back the curtain and show you the man working the levers and knobs.
To me, this is why one bass would feel different than another. Apart fro the obvious stuff like neck with and thickness, you have fingerboard radius. You can find that info with the stats like scale length, etc.
What that amounts to, is fingerboards, can be more arched or more flat. Fender typical is 9.5 radius, which would be more arched. In fact if you look at the strings going down the neck, you might even be able to see that the the strings on a fender bass are higher in the middle (A&D strings) and lower on the outer 2 strings (E&G). At least if it's well set up it would be that way. Which is a good segue to the next thing; setup. If a bass has really high action, the strings are not as close to the fingerboard as they could our should be. If everything is setup on the bass the way it should be, it will be as easy to play as it can be, and will be more comfortable in general. I've tried basses in music stores that were set up pretty badly. You just have to imagine what it would feel like with a good setup and a nice set of flatwound strings (if flats are your thing)...
So, you might have guessed that larger radius values indicate a flatter fingerboard. I play a bass regularly that has a 16" radius. It's very flat.... So, that, in my opinion is where "feel" comes from. Apart from the obvious scale length, neck width and thickness, the radius plays into the feel as well. It's conceivable that someone with small hands can prefer a very chunky neck if it has a smaller fb radius. The inverse is true as well I'm sure.
You still have to play everything you can touch that 's in your price range, but hopefully this will help you start to figure out why you might prefer one bass over another. Good luck. I hope Santa brings you something nice!!