Well that's why it's the Dollars Trilogy instead of the Man With No Name Trilogy. Even then, all those are generic names that others give him. Monco ("one handed," clearly as much of a conferred nickname as "Blondie") doesn't just favor his left hand for everything but shooting, he keeps his shooting hand in reserve. As an older and more experienced gunfighter, he's learned to keep his shooting hand ready.
I could argue that his right hand isn't good for much but shooting after the Rojos tortured him, which is why he keeps it in that brace, but I'm not sure the instant playback of the torture would support me in terms of which hand gets injured in that scene and I won't expect the leeway to chalk that up to a continuity error between movies.
Making distinctions between what the character "is" in each movie is the weakest part of your argument. In fact it's one of the motifs that binds the Dollars Trilogy more strongly than any considerations of continuity. Principal characters slip into new identities as it suits them over the.course of all three movies. Or do you believe Angel Eyes mistreats prisoners out of a genuine anger over things the South has done? That's just how these men operate.
The mercenary and opportunistic nature of the Eastwood character also distinguishes him from Harmonica, who is motivated by a personal vendetta and has more in common with a Mortimer, or a Django. The conventional categorization of characters in Spaghetti westerns as being a "Good," "Bad," or "Ugly" is really barely applicable to OUATITW. Angel Eyes would never have felt compelled, as Frank does, to a final confrontation with no material reward.