And it is my understanding that, as the early Muslims encountered the Christians, the Muslims were percevied as a new sort of Arianism by many.
I'm not too sure. Nikephoros' description is not really paying attention to religious claims of Arabs, and call them "pagans" for exemple (as for conquered provinces' population, Islam not being entierly structured at this point, they might not have found a really well defined difference).
From what I gather, identification with Arians came later, with John of Damascus. It might be more of a rationalisation on Islam, than a spontaneous identification.
As for western Christiendom...Religious identification is barely a thing up to later times, with Islamized Arabs not being fully distinguished from classical mentions.
Not that they're not considered as outside Christianity (Saraceni can name as well Arabs than what remain of pagans or half-pagan populations), but outside that, it's generally nonsense or "something that Christians don't do".
IIRC, even IXth Adoptianism (which may have some islamic influence) isn't related to the Arabo-Islamic domination in Spain.