I'm assuming that before Muhammad, the Arabs practiced polytheism unrelated to monotheistic Judaism (particularly with Judaism's historical/mythological background).
With the advent of Islam, Arabs claimed a common history (from Ishmael) and a common theological core with Judaism. How did this come about? Why would Muhammad use Judaism as his template? Or am I misrepresenting all of this?
As /u/smackaroo said, Arabia was very diverse in religious terms in the early seventh century, with many Christians, Jews, polytheists and Zoroastrians calling it their home. Indeed, the three confederations that were the most powerful tribes within the peninsula in the sixth century, the Ghassanids, the Kinda and the Lakhmids, were respectively ruled by Christians, under the domination of the Jewish state of Himyar in Yemen, and pagans (though they had to deal with the many Nestorian Christians under their rule). The Jewish kingdom of Himyar was long gone by the time of Muhammad, but it is notable that three of the clans in Medina (Muhammad's first power-base) were Jewish. These clans were exiled/enslaved in the following years for political reasons, but the initial settlement between Muhammad and the Medinans, the Constitution of Medina, effectively treated the Jews as equal members of the Muslim community. As for why Muhammad was inspired to create a new Abrahamic religion, I don't think there is a definitive answer, simply because of a lack of evidence, but from the context, any religion emerging from this part of the world must surely be influenced by Judaism, so Jewish influence on Islam should not at all be surprising.
My current view on this is hovering between 'Islam is a new Abrahamic religion that is tolerant of other Abrahamic faiths' and 'Islam was initially an ecumenical movement that was inclusive of all Abrahamic faiths from the start'. Contemporaries drew similar conclusions and had the same debate, as we have a Byzantine anti-Semitic propaganda leaflet and an Armenian history that both referenced the rise of Islam as akin to a Jewish messianic movement. Evidently, some people saw the initial conquests as something like a Jewish crusade, rather than a completely new religion. It has to be said though that even the propaganda hinted that many Jews disagreed with this view, for in the pamphlet's words 'the prophets do not come armed with a sword'.
Let me know if you have any questions :)
This is not an answer, just a comment on the presumptions of the question, but high numbers of Jewish Arabs definitely existed in the area at Muhammad's birth, as well as Christians. Pre-Islamic Arabia was a diverse mix of traditional polytheism and Abrahamic religions, so Muhammad definitely would not have been without exposure to Judaism and Christianity, especially as the traveling trader that he was.