While this shouldn't be a problem usually, you never know if you will encounter an overzealous agent somewhere. So longterm if you have the option of having your covidpass name match your id, that is ideal. What you really don't want is flying to say Moldova, and the border officials there don't accept the name mismatch and put you in 2 week quarantine right there, or while typically the covid checks this summer for me before taking a Greek ferry or entering a nightclub or the Monaco casino have been extremely cursory, once in a blue moon they check for id also to match names.