Yeah, Davison is regarded as the "team player" of the Doctors, as well as more prone to considering his companions as "friends" rather than "assistants", "students", or "pawns", the way other old-school Doctors did*. And it does seem like Whittaker is following that precedent. I mean look at the thread title for evidence. So if you're looking for a real old-school reference to compare her to, Davison really seems to be the primary one. Besides which, he was also the youngster of the old guard and Whittaker comes off as much the same.
* New Who Doctors tend to avoid having multiple companions at a time, but develop close ties to the ones they have.