I found it interesting the inscription mentioned in Thomas Asbridge's "The Crusades: The War For The Holy Land" describes Christians as polytheists and I was wondering how Christianity was viewed by Islam at this time. Specifically what lead to them to referring to Christians as polytheists?
Short answer: Muslims viewed Christians as politheists because they worshipped the trinity, of course Christians believe the trinity to be one god, but for Muslims and a lot of the Jewish scholars they are simply worshiping a man alongside God.
Long answer: to really understand this view one has to understand 2 key concepts in Islamic belief.
First is Tawhid توحيد : this is the single most important pillar of Islamic Creed, it is composed of three aspects first is tawhid ar-Rububbiyah that is the belief that there is only One God, One Creator
The second aspect is Twahid al-Uluhiyah, this means that only He Alone deserves all the worship.
And the third is Tawhid al-Asmaa wa sifaat that is the belief that God in unique and incomparable in His Names and Attributes.
And the second concept is worship عبادة : in Islam it is much more encompassing than simply saying I worship only one God so I am a monotheist, just as some protestants accuse Catholics of politheism for worshipping saints (which they respond by saying they only venerate them) in Islam worshipping only God means that all your love, devotion, respect, obedience, trust, supplications and much more should be directed only to God alone.
That is why Abu Huraira reported: The Prophet, ﷺ said, “Wretched is the slave (worshipper) of gold, silver, fine clothes, and garments. If he is given he is pleased, but if he is not given he is displeased.”
Source: Ṣaḥīḥ al-Bukhārī 6071
In this narration he doesn't mean these people believe that gold, silver... are gods that create and give life, rather they are objects of worship because they are loved as God should be loved and so they become their gods.
The pagan Arabs for example were not called pagans because they believed in several gods like Hindus, but because they didn't think it was wrong to worship creatures, the dead and objects besides Allah, they believed that there was only One Creator the God of Abraham but they believed they could also pray to other to intercede to Allah.
That is why Allah says to them (translation of the meaning):
"If thou ask them who it is that created the heavens and the earth they will certainly say, Allah. So don't you think?"
Christians and Jews are not called people of the Book in Islam because they are viewed as monotheistic religions but simply because have part of the Revealed books (albeit corrupted) and as such have a common ground of understanding