Wa alaikum assalam
The objective of the call is to signify the onset of a prayer routine. As long as the statement is 'true', I don't see a problem with this. This is no different from making use of any prophetic name and calling him the messenger of God in a call to prayer.
I note 2 underlying concerns from your question:
1. Making use of another name with God in spaces of worship
This is arguably pertaining to 'false deities', statements that are not true or unwarranted. There are verses in the Quran that clearly state that along with Allah (as a Supreme God), there had been common belief of 'supporting' deities.
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“If indeed you ask them who is it that created the Heavens and the Earth, they would be sure to say, "Allah". Say: "Then do you see what you invoke besides Allah? Can they, if Allah wills some penalty for me, remove His penalty? Or if He wills some grace for me, can they keep back his grace?" Say: "Sufficient is Allah for me! In Him do the trusting put their trust"”
2. Using the messenger’s name so as to imply a distinction between the messengers:
If this is the intention of the caller to prayer (muezzin), then this is incorrect. However, if this is simply a profession of truth and not intended to show distinction, then I see no problem in this.
Even though the practices may have altered, I do not believe that the early believers in the company of the Prophet would have ever wanted any distinctions to be drawn between him and other messengers as the Prophet himself would have arguably curtailed that view with clear narratives from the Quran.
As to if today's call to prayer is exactly the same as or even similar to the one at the Prophetic time, I would incline to agree that it is as something that finds its provenance/mention in the Quran would have arguably continued en masse as a practice and stabilised.
This is different from the Ahadith canons which were arguably not inked until centuries after the death of the Prophet.
I hope this helps, God willing.
Regards,
Joseph