The original text of the excerpt (from 1276b) is
εἰ δὴ τοῦτον ἔχει τὸν τρόπον, φανερὸν ὅτι μάλιστα λεκτέον τὴν αὐτὴν
πόλιν εἰς τὴν πολιτείαν βλέποντας: ὄνομα δὲ καλεῖν ἕτερον ἢ ταὐτὸν
ἔξεστι καὶ τῶν αὐτῶν κατοικούντων αὐτὴν καὶ πάμπαν ἑτέρων ἀνθρώπων. εἰ
δὲ δίκαιον διαλύειν ἢ μὴ διαλύειν, ὅταν εἰς ἑτέραν μεταβάλῃ πολιτείαν
ἡ πόλις, λόγος ἕτερος.
The translation I would favour which is relatively literal is the one given by Harris Rackham, published in Loeb Classical Library (beware the citation error in the linked Perseus' web source):
Therefore if this is the case, it is clear that we must speak of a
state as being the same state chiefly with regard to its constitution;
and it is possible for it to be called by the same or by a different
designation both when its inhabitants are the same and when they are
entirely different persons. But whether a state is or is not bound
in justice to discharge its engagements when it has changed to a
different constitution, is another subject.
The corresponding sentences are boldfaced as the one in the question.
So far as I am aware, Aristotle does not take up the issue separately by itself. I presume the reason is that it is a matter that could be properly treated on a case-by-case basis rather than by an overall judgement. Here, πολιτεία (politeia; "the form of government" and "constitution" in the given translations, but also points to "engagements" in both translations; there is no separate word for "engagements" in the original text) stands out as the key word in the focal sentence. This is a highly theory-laden term of Aristotle as well as Plato before him. In the present context, its implication as "engagements" might be taken to denote the contracts between the city as a political body and its citizens considered both individually and collectively (i.e., how the city and the citizen stand with respect to each other). That is to say, social and political contracts change along with the constitution. As the history, especially of the 20th century during which many states and political systems collapsed and new ones were established, attests, the preservation of reciprocal rights and liabilities between the political bodies and the people subject to them through the changes became causes of harsh disputes and hardships, some of which have not been settled as yet.
Hence, Aristotle seems to suggest that each case should be decided by itself according to the doctrines he propounds in his treatises on politics and ethics (also we should not forget that his views on these topics exhibit admirable coherence with his fundamental philosophical tenets). I think Peter Simpson's following view (A Philosophical Commentary on the Politics of Aristotle, the University of North Carolina Press, 1998, p. 139) is complementary and supportive of the interpretation I offer:
1276b13 From this result it now follows, of course, that when the tyrant or the oligarchy acts, so does the city. But this conclusion
does not settle what lay beneath the democratic argument quoted at the
beginning. For that argument was really about whether a democracy was
bound by the contracts of a previous regime and that, as Aristotle now
expressly notes, is a different question. How he would answer it he
does not say, but one can conjecture that it would depend on the
circumstances, that is, on whether keeping this or that agreement
would really serve the common good (which is what is meant by
political justice, 3.12.1282b16-18). At all events, the first dispute
from chapter 1, about the identity of the city, has now been resolved
and in such a way as to confirm the definition of city in terms of the
citizens or the regime.