Human moments with non-super-heroes like this are what distinguished Miller’s Daredevil from what came earlier. Daredevil was always a little different from other books in that it really focused on the intersection between super- and secret-identity. Spider-Man focused on Peter Parker, and his need to keep a secret. Batman focused pretty much exclusively on Batman–Bruce was just a thing that filled space between Batventures. Clark Kent was pretty much the same. The Fantastic Four didn’t have secret IDs. Bruce Banner, like Parker, was tormented by his super-ID.
But Daredevil really folded the two personae, even right from the start. There were those awkward times when Matt “died” and “Mike Murdock” was created, but even those stories showed a real need for Murdock to be “public.” He reveals his identity to every chick he dates. He’s clearly a publicity whore.
It’s a real cool concept, and Miller begins to exploit it thoroughly. Brian Michael Bendis would tell the definitive tale about it, but he couldn’t have done it without Miller’s foundation.