Chatting with my brother...

Me: "So, the Doctor's regenerating, but Clara's staying on."
My brother: "Huh, I don't think there's been a companion who's stayed on through a regeneration before. Except Sarah Jane."
Me: "Ben. Polly. Tegan. Adric. Nyssa. Peri. Mel. Rose."
My brother: "... Well, I haven't seen a lot of classic Doctor Who!"
Me: "Me neither."

LOL,funny.

So really, only 2-3 and 10-11 did not have companions stay thru....not counting the 7-8-9 period. Stay-thrus are more common.

Well I think that when it's the same companion staying on, it adds to the feel that things have changed. The companion would have to deal with the fact that they're looking at the same man in a new body, which adds to the awesomeness of it all. If it was just a new companion as soon as he regenerates, they wouldn't have known the difference. So I think that's probably why stay-ons are more common.

It shows things changed, but at the same time, it is the same. Like you said,same man, different body. And same show. When 11 came on, it felt like a new show. Everything was different.