Yes, it is the same Hulk character, same universe.
Hulk (2003) had a sequel in The Incredible Hulk (2008), which was also a sequel to Iron Man (2008), thus it was a crossover between Hulk (2003) and Iron Man (2008).
But the 2003 movie is not considered part of the Marvel Cinematic Universe (the MCU) because that's Iron Man's universe (synonymous with the MCU). So it's two universes crossing over in The Incredible Hulk and most of the following films and series that share characters from Hulk and all characters introduced in the following crossover installments.
Watch both films and notice how The Incredible Hulk picks up from where Hulk left off. The newspapers/articles/documents shown in the opening montage also explain what happens in the time gap between the years the two films take place in, matching that time gap and the narrative.
Marvel Studios created a movie studio & wanted to create a cinematic universe & went all in with Iron Man. The success of Iron Man determined if they'd float or not. They also collaborated with Universal & made The Incredible Hulk at the same time, a sequel to Hulk, but in Marvel's own style, latching on to a movie universe already established (Hulk).
There were creative tensions when making the film because Universal wanted it more as a sequel to Hulk and Marvel wanted to make it more of their own.
There were also tensions between Edward Norton, alongside the director Louis Leterrier, against the studio, Marvel. They wanted it more character driven - like Ang Lee's Hulk - whereas Marvel wanted a more action-oriented quickie.
There's a finished cut of the film called the 'Norton Version' that was supposed to be the released version (almost an hour longer), but Marvel cut it down, significantly, right before release. (It's hard to find, but it's floating around thanks to piracy).
Leterrier describes the out-takes: “It’s like the back story. It’s more the sequel to the Ang Lee movie.”
In an interview with Gale Anne Hurd, producer of 2003's Hulk and 2008's The Incredible Hulk:
Collider: It's set up so that it could be a sequel or a remake, depending on how you look at it. Was that something you were very conscious of?
Gale: To be honest, that was was one of those things where we said, "Look, we don't want to tell another origin story." One thing we absolutely knew for certain was that we didn't want to start with Bruce Banner and how he becomes the Hulk. From that point on, we couldn't quite figure out how to term this. I think at one point we asked if anyone could come up with the appropriate term and someone one line had a "re-quel" which I think was actually perfect. It's kind of a reboot and it's kind of sequel.
Iron Man opened May 2008 to unexpectedly enormous success, while Incredible Hulk opened one month later to tepid success, which, creatively, had one foot in Hulk universe & one in MCU. Marvel doesn't own 2003's Hulk (movie), so deny connections to it.
Marvel Studios owns the character of Hulk, but Universal owns the distribution rights to a Hulk film. That's why Hulk appears in Marvel films but we have never seen another solo Hulk film, because Universal would be needed to distribute it. Marvel & Universal realized they don't play well together, so there's never been another solo Hulk film since.