Apart from the fact that he first met River as Ten, and we’ve had references to Rose and Martha and Donna in the show since he regenerated.
That said, what I think is important for NuWho fans to understand, whether you’re willing to watch Classic Who or not, is that Ten was the exception, rather than the norm. The longing for a past companion, constant references to what came before, and certainly to the previous Doctor are traits that were pretty much exclusive to Ten. Things haven’t changed with Eleven, they’ve gone back to the way they were.
I suspect the reason that Ten was different is that we only got one season with Nine, and that was because Eccleston wanted to leave, rather than his leaving being the plan all along. I could be wrong, but I’m sure that a lot of Ten’s first season was originally planned for Nine, and that’s why it felt like there was such continuity between Doctors. Ten’s departure was planned for, and so it followed the pattern of previous regeneration’s more closely. While, yes, there have been companion overlaps before (Sarah Jane, Peri…), once those companions were gone, they were gone. The Doctor has travelled with so many people, and they are all in his hearts, but if he spent time dwelling on them all it would break him. Doctor Who is a show about change and progress, it moves ever onwards.
As I said, Ten was the exception to this, not the rule.