Because those other members of the Organization didn't bother differentiating themselves.
Roxas woke up without memories of his past life and they never came to him. He was a clean slate and was able to forge his own identity based solely on his experiences from creation on. The other members of the Organization remember their past lives and even though they make a point to say that can't feel (which ends up being false down the road) they "remember what it's like." Their entire identities are formed on the lives they lost, and so because they hold onto a sense of continuous self, the hearts they regrow are indistinguishable from what they had before. They're just ignorant to them.
To put it more clearly in a context where this is spelled out for us, let's look at Axel. When Roxas asks Axel what he was like before he became a Nobody, Axel says he was more or less the same guy that he is now. When he reforms as Lea, he has memories of his time as Axel and though he starts off as 'Lea' eventually he asks to be called Axel again, the line in the chain never breaks -- there is no chance for a new identity/heart to spawn.
They don't see the differences and their lives are continuous existences.
Xemnas, however, did form a unique identity given the multi-stacked heart situation that lead to his creation. He renounced "Xehanort" as an identity and formed as someone unique and new -- a special Nobody. In instances like this where there is enough of a difference that the person reclaims a new sense of self, then you're looking at a different person.
The Organization's real goal, as explained by Xigbar, was to get the members to renounce that sense of self so that Xehanort could more easily overtake the heart and literally just make everyone into a Xehanort clone. However that's not so simple once the person has a lived experience or if they had memories of who they were before.
In short: in instances were Nobodies consider themselves to be a splintered part of a previous existence that they can remember, that's what they are; because their sense of self is shaped by that life, that sense of existence just continues and the heart that recompletes merely absorbs the experiences of the prior life.
In instances where the Nobody forges an identity that is not contingent on or continuing the life they had before, it creates a new personhood. Even if that Nobody recompletes, the heart they grow is not an extension of what or who they were before and so it is separate, though the two remain linked.
They see themselves as unique, and so that's what they become.