No automatic conclusion follows from the name alone. Identify the actual legal and employment change and have its effect on the immigration matter assessed.
Distinguish appearance from legal substance
A rebranding may differ from a transfer to another entity. Explain the actual structure, ownership or control changes and employment facts. Familiar duties do not resolve the company relationship question by themselves.
Work out what a reader would have to accept for the answer to be that nothing needs to happen. They would have to accept that the applicant's employer is the same legal person as before, that the qualifying relationship with the foreign entity is intact, and that the approved duties are unchanged. Each of those is checkable, and checking them is faster than debating the significance of a new logo.
Where all three hold, record who confirmed each and move on; that record is what makes a later verification uneventful. Where one fails, the file has a specific issue with a specific owner rather than a general concern about rebranding. The trap here is the change presented internally as cosmetic because nothing visible moved: the same building, the same team, the same customers, and a different employing entity on the payroll record.
Visible continuity is evidence of very little on this question.