Is it possible to flag a User, rather than just flagging all the spam answers that user posts?
Kind of, but not directly. You can't flag the user, but you can flag any (rather than all) of the user's posts with the other (needs ♦ moderator attention) reason and explain the situation. The mods can take it from there.
If you encounter a user posting nothing but spam, and it's clear the user created that account simply for the purpose of posting spam and nonsense, it's best to flag as spam. Although, as Jim says, feel free to also flag a post as "other" if there really is something you need to describe in greater detail, but in general spam needs little explanation.
The Stack Exchange system is programmed to handle these types of flags differently than other flags. They stand out a lot more, and there are measures that Stack Exchange put in place to help prevent more spam from that same user.
When I see a post flagged as spam, I generally do some digging to see if the user has a pattern. Hope this helps!