You're close, but some Christians would argue that the god worshipped by those of Jewish faith is not the same god either and therefore not embrace that god. Those Christians would say that since Jesus revealed the trinitarian (Father, Son, and Spirit) nature of their god, to reject that nature is to worship a different god altogether. Similar to how Muslims acknowledge their shared history and feel a respect for Judaism and Christianity, those Christians accept and respect those of Jewish faith, but will still point out their incomplete understanding of the god the Christians worship.

Christianity embraces the God of the Torah but rejects the Muslim faith.

Still, Allah is the same being as the Christian "God".

I'm not saying Islam is canon to Christianity. Just that when Christians talk about God and when Muslims talk about Allah, they are talking about the same being.

Just like in English, we call the protagonist of the Pokemon anime "Ash", but in Japan, he's called "Satoshi". But it's the same character no matter which name you refer to him as.