Let me start out by saying I have very little understanding of the mechanics of a Bass Guitar. I'm an amateur who plays as a hobby.

Now that's out of the way, here is a picture of the the top pick up the one closest to the fretboard.

Here is a picture of the bottom one, farthest away from the fretboard.

The screws seem loose in the bottom one. I can actually push it into the body a little bit. I'm wondering if that's how it's supposed to be? Cause I've had this bass for two or three years now and I'm fairly sure it's always been like that.

It's a Fender Jazz copy, so if it's messed up that's probably why.

If it should be screwed back in, is it a simple enough fix that I could just tighten the screws?
Those screws let you set the pickup height.

If you don't like them being that high, just tighten them up till they're set to a level you like.
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Usually there'll be a spring under there to let you adjust, mine are like that aswell, it's no big deal. If you really want you can adjust it back but that's only for looks.
In a lot of basses there's actually a piece of foam under there, instead of a spring. The concept is the same though...your pickup is fine. If the screw is sticking out of its hole, you can pull up on the pickup itself to put everything in the proper position.
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