Well, of course, Messianic Jews/Jews for Jesus do regard Jesus as the incarnation of God, and also consider themselves to be Jews. Mainstream Jews, however, disagree with them on the latter point.
As others have pointed out, Judaism involves prohibitions as well as positive injunctions, and a huge prohibition is “idolatry”. Idolatry involves worshipping as God anything which is not God. The mainstream Jewish concept of God entirely precludes the possibility that he could be human, or incarnate. Therefore, to worship Jesus as God is idolatry. Therefore, it’s not Judaism.
A Jew embracing Islam would not face quite the same problem, since Muslims worship God and have no truck with any of this incarnation stuff. Still, what FinnAgain and DSeid say looks right to me. Embracing a religion which is not Judaism is, well, not Judaism.
And, remember, in this context embracing a religion is not really about what you believe, but about what you do and how you live. A Jew could certainly adopt a variety of beliefs which are shared with Christians or Muslims or some other group; that’s not necessarily problematic, except in so far as holding those beliefs might be inconsistent with observing the Law. What’s problematic is joining the other group; sharing religious practice with them; living as a member of that group.