I swear there was a theory floating around a few years ago re: The way the Doctor tends to regenerate, according to the influences pressed upon in him in his last incarnation. These influences tend to come from his companions.
I think it was this, at Tor.com ?
Moffat and RTD both mentioned that theory a couple of times. It lines up with some of the Doctors, not all of them. There are similar theories going around saying that, for example, the nature of his death and his last adventure shape the new Doctor.
The only real truth in that doesn't lie in the 'lore' (as far as that exists in Who), but more that every Doctor and the tone of every era is a reflection of the times and the rest of the media landscape, and a reflection of what came before. Writers, even new ones coming in, expand on small ideas that worked, dismiss stuff that didn't or that have been mined completely.
If, say, Moffat would have continued the show in the 80s and written the next Doctor after McCoy, his first Doctor would have been very different than Matt Smith, who was (even though he was his own man), very much a reaction to Tennant. And Capaldi came from Smith - if Smith wasn't his predecessor I'm sure the tone and the way he played to role would have been very different.
TL;DR - it's a fun theory but more something that's applied after the fact. Not really an in-universe thing. Like most of these, it falls apart if you look too closely at it and try to apply it to every Doctor.
Not trying to be a Debbie Downer though, if anyone loves mindlessly philosophizing over this show it's me.