I know, for instance, that Muhammad's tribe was in control, in full or in part, of the pilgrimage trade in Mecca. The adoption of a pilgrimage to Mecca by Muslims can clearly be seen as a (perhaps financially-motivated) continuation of this tradition.
Yet at the same time some Islamic scholars rejected the depiction of living things in art, or adhered to aniconism, which is clearly an intentional departure from Christian / Manichean artistic tradition.
So what are some of the other continuations / rejections? What explains the origins of some of these?
Super interesting question.
A quick point of clarification before I dive in: it is true that Muhammad's tribe controlled the pilgrimage and trade, but it's rather unlikely that this was co-opted into Islam for financial reasons. For one, the revelation for Muslims to participate in the pilgrimage to Mecca came while non-Muslims were still firmly in control of Mecca and basically at war with the Muslims of Medina, pre-dating the Meccan conquest by at least several years. At the treaty of Hudaibiyyah, Muhammad signed away significant rights (to the great consternation of his closest advisors) in order to get Muslims unobstructed access to complete the pilgrimage. Moreover, revisionist historians have questioned the actual importance of Mecca as a trade center, given that it doesn't seem to be mentioned as such in other sources of the time, which notable economic centers would have been. It's not unreasonable that later Muslim historians would've overplayed the significance of Mecca, given its later role as the spiritual capital of Islam, as well as the site of Muhammad's greatest military triumph (the unconditional surrender of Mecca).
To your question: the short answer is that much of Islamic rituals and symbols are co-opted from other belief systems. Much of this was done consciously; Islam claims heritage from other Abrahamic religions, and its initial rituals were often developed relative to Judaism and (to a lesser extent) Christianity.
For example, in early Islam, Muslims would pray towards Jerusalem, until Muhammad received a revelation to turn towards Mecca mid-prayer. The Muslim holy day and congregational day, Friday, was likely chosen in reference to the Jewish Sabbath on Saturday, in order to not overlap, but be close enough to tie the beliefs together.
There are some theological bits and pieces borrowed from the polytheist faiths of Mecca. For example, Arab polytheists did in some cases worship Allah, although Allah was viewed as a rather remote supreme God, rather than a personal one who would actually be likely to respond to your prayers about, say, a good harvest.
You've already mentioned the pilgrimage to Mecca as a similarity. The Torah requires a tithe from crops to be paid, while Islam has the Zakat tax.
Of course, there are significant differences. The Muslim prayer ritual, Salat, is highly unique; the frequency of prayer is also much higher than in other religions. (Islam's explanation for this is that Muhammad was taught how to pray directly by the angel Gabriel). Fasting is mandatory and highly structured (no other religion that wanted adherents would've tried this seriously in the Arabian desert). And even where there are apparent similarities (charitable tax, pilgrimage), significant innovations have sprung up around these rituals to make them highly distinguishable from pre-Islamic versions.
Here are some additional sources if you'd like to read more:
El-Islam, M. Fakhr. "Arabic cultural psychiatry." Transcultural Psychiatry 19.1 (1982): 5-24.
Secularization in Multi-religious Societies Indo-soviet Perspectives, by Dube and Vasilov (skip to Chapter 17)
The Development of Islamic Ritual by Gerald Hawting
No God but God by Reza Aslan
Mecca and Eden: Ritual, Relics, and Territory in Islam by Brandon Wheeler