No, you're wrong on that one. If you search through GTA 3's files you see the name Claude. Claude said that his surname may or may not be Speed. So even if they are different entities, the III counterpart was not by any means meant to be nameless.
The likely hood of it being the same character is high, but the bigger factor regardless is the era system.
I stand corrected, then...
I still don't like to think that Claude Speed from GTA 2 is Claude from III, though. GTA 2 takes place in the future, like 2020 or something and III was modern for it's time, at 2001. That's almost a 20 year difference from GTA 2 to III, but it's forgivable. However, when you take into account that SA took place in 1992, then that spans like 30 years and that's my problem with Claude being the same person in 2 and III - he would be too young in III or too old in 2. But his age is never specified anyway, so maybe he is 50 years old or something in GTA 2, doesn't really matter i guess. I just don't like to think they're the same... Either way, as you said, the 'Eras' kinda f*ck with this entire ordeal anyway.