The C++ standard requires:

  1. Both std::vector<T> and std::basic_string<T> support push_back(T) with amortized constant performance.
  2. Additionally, std::basic_string<T> has an operator+=(T) that behaves semantically identical to push_back(T), but does not have a complexity requirement imposed by the standard.

Logically that leads to every reasonable standard library implementation to simply dispatch std::basic_string<T>::operator+=(T) to std::basic_string<T>::push_back(T) (or vice versa, of course) and have both operations run in amortized constant time.

You're technically correct that the standard theoretically allows push_back(T) and operator+=(T) to have different time complexities, so you could make operator+=(T) run in linear time if you're trolling (but in that case, why stop at O(N) and not make it O(2^N) or something?), but since push_back(T) and +=(T) need to be equivalent and the former needs to run in O(1) amortized time, there is no reason to make the latter perform worse than the former.