Does the PlayStation 3 have the compatibly to play PlayStation 2 games like how the PlayStation 5 does for the PlayStation 4?

Only the very early PS3 models that came with 20, 60 or 80 GB hard drives and had 4 USB ports on them can play PS2 games. Not even all "fat" PS3 models can. Those are also the models most likely to break down (YLOD or laser failures) due to their age.

Original fat PS3s play ps2 games, the newer fat ps3s and slims no. 

Only early models can.

"Yeah, sure." - Nicolas Cage after reading ANY screenplay