Yes, many countries expect or require their citizens to enter the country on a passport of that country. Not all, though –.the UK and New Zealand do not, for example. The UK, for instance, merely requires that the documents presented at the time of immigration are valid for the intended purpose of the visit/stay.

All of that aside, whether another country would recognise that Bill Smith or Mary Brown is the same person as the newly re-named Bill Jones or Mary White would depend, I suppose, on their databases and how flexible they are.