Why would a potential supervisor emphasize kind of doctoral degree?

I'm guessing that he has had a number of candidates who were unclear on that from the outset and that he's simply clarifying to make sure there's no confusion prior to your joining the lab. I don't know whether this is likely to affect your long-term career, hopefully someone who knows more about German academics can clarify in the comments.

Whether a German department of mathematics grants the title Dr. rer. nat. or Dr. phil. has nothing to do with the research done in the institute or the difficulty of obtaining the degree, but is the result of some historical happenstance. At most universities departments of mathematics are quite young, and they may be split of an older physics department or a philosophy group, in which case they would have just inherited the naming of the title.

The reputation of Dr. rer. nat. and Dr. phil., as long as they were achieved in the same area, are the same. Dr. Ing. is somewhat different, but not easier. Dr. med. is not taken seriously. There is also the Dr. hum. biol. (human biology), which can be obtained by students of other subjects than medicine, and is on a par with Dr. rer. nat., i.e., a Dr. med. for someone who actually did research.