Interesting discussion. A Jew can ONLY be a Jew according to Jewish law - anything else is forbidden. Truthfully it depends on who you hold by though in regards to Islam and Christianity, also. Some hold that Christianity is actually ****tuf. ****tuf is hard for me to explain as I don't know the laws of it so well and exactly how it applies, but, basically the general idea as I understand is as follows. Effectively they do believe in the same G-d as us(NOT JESUS), what they might call the "Holy Spirit" or however they decide to coin it. But, the idea of ****tuf is "divine partnership". It is absolutely forbidden for a Jew to take part and belief in ****tuf, but for Gentiles it is acceptable. The idea of this Jesus fellow being a deity of some sort in lesser flesh that is partnered with the one and only G-d, isn't according to some opinions isn't something entirely forbidden to Gentiles. That doesn't mean it's right for them to believe in it or acceptable necessarily though. It's a very disputed subject.
In Judaism there is large disagreement amongst the rishonim(the sages of around 11th century to 15th) in regards to decidine whether Islam and Christianity are acceptable. It's still unclear today depending on which of the Rishonim you hold by. I personally agree that Islam is definitely monotheistic and it's possible they believe in the same G-d as us. That I won't say for certain. Christianity could be considered idolatry depending on what movement you're talking about. Coptic Christianity definitely is not idolatry, but you could say that much of western Christianity is. Your question about Islam though still isn't necessarily resolved even to this day. But, to my knowledge most orthodox Jews hold it is an acceptable form of monotheism for Gentiles, not for Jews. In fact, the Rambam(Maimonides) - perhaps the most revered sage of all time holds that you technically could pray in a mosque if there isn't another acceptable place to pray around. Whereas a church is utterly forbidden to pray in, or even around(I.e. In front of it in the open; etc), so much so that according to the Rambam we are forbidden to even bow down in front of it as it may appear to others we are praying.
It's interesting, many of the Rishonim who come from the east consider Islam to be more idolatrous and Christianity more monotheistic - and those from the West consider Islam to be more monotheistic and Christianity to be more idolatrous. Which sort of makes sense geographically if you compare western christianity to earlier eastern chrstianity. Depends who you hold by on this subject. But most Jews I've met agree Islam isn't idol worship.*
