Can you use this phrase ?

Mark is doing an apprenticeship as an engineer.

Is the expression doing an apprenticeship possible, or do I have to use serving an apprenticeship?
Dear Stephan

"He's doing an apprenticeship" is fine in UK English

"He's serving an apprenticeship" is OK. It is proper, but is maybe now a little old-fashioned

In common speech, you could say "He's on an apprenticeship"

Regards, Dave
It's the same in AmE.