There are some airlines where a multi-segment flight with the same flight number does stay on the same aircraft through all of the segments, but that is not true for all airlines. Where it is not true, there is the usual risk of missing the connection and the like.
I used to train Reservations Sales Agents at Disney World. We sold packages that included air. The difference between a nonstop and direct flight (aka multi-segment with same flight number) was incredibly difficult for some people to grasp. No matter how many times we covered it, the majority of people still got it wrong on quizzes.
Part of the curriculum was to mention what you highlight above. Namely, that it's possible to miss a connection on a direct flight because the airline might operate each segment on different equipment. All the trainers got together and decided to skip that part of the lesson. It just confused most people and it wasn't something the agents would ever explain to guests. (If a guest experiencing IROPs ever called, they were immediately transferred to the Air Desk.)