I've had multiple that do it. they're quite visible in the right light. you can also see it when your yellow cartridge runs out, despite printing exclusively black and white, while the other color ones are full.

"most" might not, I can't claim anything there and EFF claims most don't, when they were checking on it in 2015-2017. I'm pretty darn sure that some do though.

As far as I know, the most common form of tracking in inkjet printers is in the ink itself. Companies add chemical markers to their inks which can show when the ink was made, and which company made it. The FBI collects that data in the International Ink Library https://en.wikipedia.org/wiki/International_Ink_Library

There are also databases cataloging the specific printing characteristics of various models of printers and research has demonstrated that it's possible to trace documents back to individual printers based on slight variations in how they print.