I've ranted on this one before so I'll just leave you with this...
UNIVERSE ONE: GTA, GTA 2, LONDON 61, LONDON 69
UNIVERSE TWO: GTA3, VC, SA, LSS, VCS
UNIVERSE THREE: GTA4
"Rockstar president Sam Houser—discussing the development of GTAIII in an interview with Edge magazine—said:
|"...there’s been a lot of debate about these things—like whether [the GTAIII lead character's] name is Claude Speed, or whether he’sthis or he’s that—and it was a lot less planned out than that." - Edge magazine (2008-03-16).|
His name was in the code but the code isn't used in the game and the reason is because the original idea was to let you name the character yourself, Claude was the DEFAULT name in homage to GTA2.
A quote from WikiGTA:
|"Claude Speed is the protagonist of GTA2, as mentioned in the credits and manual of GTA2, as well as in GTA2: The Movie. It is possible Claude is based on this character."|
And a few more points...
1) Some differences are notable, such as slightly different clothes and another hair color
2) Claude does not speak in GTA3, but Claude Speed does in GTA2: The Movie
3) The name 'Claude Speed' was never used in the GTA3 series of games. The name 'Claude' wasn't actually used ingame until GTA San Andreas, probably as a homage to the fans.
4) Claude hasn't aged in twelve years if GTA2 is set in 2013
5) GTA2 and GTA III are not set in the same canon.
So, no, they are not the same person but as has been said earlier, Claude may be an 'alternate universe' Claude Speed.