I did some reading on this a couple of years ago. I still want to paint ours!
I wish I could remember where I got all the info, this is what I remember: The first problem you might have is cleaning/prepping the floor. As I recall
all traces of grease and oil have to be removed from the floor
completely or you'll have paint lifting. There's some kind of acid wash you have to apply to really get it all out too. Unfortunately oil/grease seeps into floors over time, so if it's an older floor you may never be able to get it all out and over time it will slowly lift any paint you put over top of it (which will then crack, peel off, etc). In some areas moisture will penetrate the concrete from below and lift the paint from underneath too. That's what I read anyway - your mileage may vary.
There are garage paint kits available in the stores like Lowes. Rustoleum markets an Epoxy floor paint that has a contrasting flake you sprinkle into the paint as you apply it. The sprinkles give it some "grip" so you don't slip on the floor too I think.
You can check the Rustoleum website for info on their products, but I also found this one with some general information on the subject:
https://www.epoxyproducts.com/garage4u.html
As for the other flooring material - I would think if you prep the concrete for paint your tiles will lay down fine. I'm not sure about linoleum (or whatever material they're commonly using now) though. Certain solvents might eat away at the tiles, and if they're actually
tiles anything you spill (oil, coolant, etc) will end up seeping between and then under the tiles causing them to lift over time. If you're prepared to pay the price you might consider the large open-cell tiles they sell for garages and racing pit-areas. They're not cheap, but they're very sturdy, interlock with each other (no glue necessary, they lie flat on their own), and you can easily lift them up from time to time and clean under them. They come in a variety of colors so you can make a pattern under the car if you want (checkerboard, a border around the outside, etc). Places like Griot's Garage and California Car Cover sell them but shop around.