>Game Boy version

So what I was wondering about was not correct but it was worth a look (the JP version still says he's Dracula son).
This is where I believe they intentionally changed the character's story. There is no more claiming "I am Dracula", no more sitting on the throne at the end, and the in game story makes several mention of his father, while the Famicom version had none. In other words they made a point of removing anything that says "I am Dracula" and instead adding several mentions of his father instead.
The manual says he is the son of CV3's Dracula, though that still doesn't say he's CV3's Alucard (again if they wanted it to be that, why wouldn't they say so? I'm sure they considered it but since Alucard is meant to be fight his father it didn't fit).

This was done 3 years after the Famicom version by a completely different dev team, my theory is that they changed the character's story because this time they had an international release and the concept of SD characters just wasn't known in the west at that time, I doubt even many western anime-fans would know.

Anyway fandom-ism likes to pretend that series have *a* canon and that everything is supposed to fit together together like there was a grand plan from the start, but I don't believe that's how it works and it often leads to reaching to make the theories stick. I like to view works on their own or at best in relation to their immediate sequels or predecessors if they are connected.
Umbrella in Resident Evil (1) was not the same Umbrella as in the sequels. Kyle Katarn is still alive. The princess of Moonbroke does not have a sister. Dracula-kun on Famicom is Dracula.