According to Jewish law, if your mother is Jewish then so are you. Since he has yet to convert we can say with certainty that he is.
That said, is he Jewish like Jackie Mason? I don't think so. He's Jewish like Karl Marx, the father of Socialism. Not exactly a Jew to be proud of. FYI, Karl's Jewish father converted to Christianity because he was scared of anti-Semitism. Later Karl was baptized as a Christian as well.
Sadly when I confront young Jewish adults about their love for Bernie the argument goes something like this:
1. I show them a video where Bernie condemns Netanyahu for budding into US politics when he spoke to Congress about the Iran Deal. I title it: Bernie is no friend of Israel.
They answer: Just because he doesn't like Netanyahu that doesn't mean he doesn't like Israel.
2. I send them an audio of Bernie condemning the IDF for indiscriminately murdering 10,000 innocent Palestinians.
They answer: Silence.
3. When Bernie gets heckled by a black man accusing Jews of running corporate America they say: You see, he answered that he's proud to be Jewish.
I respond: He never called out the man as being an antisemite, instead out of nowhere he makes a moral equivalence statement between the Palestinians and Israelis.
4. Finally I confront them face to face and call Bernie out for being anti-Israel and for distancing himself from all things Jewish.
They answer: Just like when you (me) criticized Obama for attacking Israel, Israel still stands. When we vote for president, issues pertaining to Israel are way down on my list.
Finally the truth comes out! I would respect the young foolish Jewish adults a little more (and I do mean only a little bit) if they would respond with the response for # 4 above in the first place. In other words, rather than defending Bernie as not being a self hater they should simply admit: I love all the free stuff we're gonna get from him and I really couldn't care less about Israel or Jewish issues.