What type of paint can I use to cover this plastic kitchen tile?

I would remove the plastic tiles and put up something else....maybe pallet wood, or I've seen people use vinyl plank flooring as a kitchen backsplash.

I’d replace them with ceramic tile.

ahhh my first house..metal cabinets ,plastic tile. getting the glue off the plastic was horrible, I opted for a cover up, a very thin dry wall board went over the tiles.

They haven’t made plastic tiles in 45 years, so no point in trying to patch them.

When putting ceramic tiles on a wall that is prone to water splashing & needs serious cleaning for food-splashes, I would not use thin-set. It is not water resistant.

After removing the plastic tile, I would sand everything smooth, Prime with mold-killing and stain-blocking primer and then paint with an exterior grade latex. We are trying to save your old wall boards.

Then I would use a petroleum based mastic on the wall for the tile’s adhesion.

Then I would acrylic or polymer additive to non-sanded grout & use pennies for spacing the tile.

If you use dot-matted porcelain mosaics, the grout joints will be already done.

It looks as though the tiles are beginning to become unstuck from the walls. Instead of spending money trying to update them, it may be time for the whole lot to come off, and to redo the entire wall and behind the sink properly.

Remove the plastic tile and replace with ceramic. Not worth the trouble to repair and/or paint.

https://www.hometalk.com/diy/tile/walls/learn-to-tile-like-a-pro-22406597

https://www.hometalk.com/diy/decorate/walls/easy-inexpensive-subway-tile-1476167