If your CD player had cheap anti-skip it probably does lossy digital audio compression before output. A lot of the CD-player-as-a-package chips had older, crappy lossy audio compression and saved to a small bit of RAM on the CD player. Not much of a power envelope for compute power for audio compression logic. With memory being really expensive back in the day and prices being cutthroat there wasn't much memory for the blanti-skip buffer. So you needed fast, cheap, and really compact audio compression. Nobody really bothered improving it once MP3 players came out and memory got cheaper, so even "new" CD players use the same hardware portable CD players were using in the 90s.

And even then, it's not digital square waves coming out of your headphones. At some point that digital signal needs to be converted to analog waves. The quality of the DAC matters as well and can give a different quality of output.