I'm 99% sure that anecdote is made up. The only source for it is a book on modern art history by a guy called John Baskett who as far as I can tell isn't a professional historian.

The art styles of the two eras are completely different and the two Emperors look nothing alike. Sculpture from the era of Marcus Aurelius is mostly an attempt at realistic depiction, while a sculpture from Constantine's era is mostly, but not universally, more stylised. Both Marcus Aurelius' and Constantine's likenesses are well attested in various forms of art and these depictions of them are very consistent. Marcus Aurelius is depicted with thick, curly hair, a full beard, small eyes with very heavy lids, and a sullen expression. Constantine is depicted with short, cropped hair, being clean shaven, huge wide-open eyes, and as having very heavy features with a large chin and jaw. They could hardly have looked more different.

On top of that, it's not like Marcus Aurelius was an obscure figure to Medieval Christians. He was held in very high regard and was thought of as a great philosopher. I can't imagine why they would want to destroy an artistic depiction of him in the first place.