I understand AはBとする as "A assume/regard B."
You have the right idea but not quite the right implementation.
Let's use the word 'regard'. This word takes three components: X regards Y as Z, where X is the subject, Y is the object and Z is the -- whatever you call that in English. So the Japanese is Xは/がYをZとする. The thing that is getting regarded is Y so this makes sense as the object, and using と makes perfect sense here since it is a particle of comparison.
In your sentence the subject is 人, which is promoted to the head of the relative clause so we have
People (X) who regard Japanese (Y) as their mother tongue (Z)
I deleted this answer because I thought that maybe using the word 'regard' was a little misleading. I think XをYと思う would be more literal, but since you seem confused about the assume/regard angle maybe I should try and say some more.
XをYとする literally means make X into/as Y. So I think "regard X as Y" is a reasonable translation. When it comes to 'assume' I can think of two meanings and I'm not sure which one you are thinking of. The first meaning would be 'to make a judgment without evidence'. I don't think that fits here. The second would be 'to take on a role/character/etc'. I guess 'people who take on Japanese as a native language' kind of fits here but it's a bit awkward. I think it is easiest just to think of the literal translation, 'make X as Y' and mold it to the context using your native language.