Quite often they will be. For instance imagine some debugger which is trying to print a backtrace in a useful way. It's going to want to map between the procedure objects sitting in the backtrace and their names. And that map is either going to point at the 'wrong' name, or it's going to point at all the names and you then have to know which one you actually used.
Here's an example in Racket:
> (object-name cons)
'cons
> (define make-thingy cons)
> (object-name make-thingy)
'cons
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