Germany is complicated in this way, to be fair ... "Greens are not a natural Israel ally" does NOT quite do this justice ... and I am saying this as someone who, for professional reasons, works with a lot of Germans (and also has had a lot of, sometimes heated, discussions with them over politics ... I also speak really fluent German).
Okay, this might get long-ish - propbably a short essay - and also: complicated. And, by necessity: incomplete and not doing full justice to all aspects of it - but it needs explaining:
First things first: you need to understand what happened to Germany after WWII. Which, and please note that I am absolutely and grossly over-simplifying here, is basically "not that much at all", also "everything". Germany was split in half, basically divided by Western and Soviet occupying powers, after WWII. And while each side did its own bit of "reckoning", each in its own ideologically acceptable manner, the real debate about "but we were the Nazis" - having begin in roughly around the 70s but not having gained major traction while the cold war was still going on - really only happened after German re-unification in the 90s. At which point, it sort of did in earnest and became a major thing. Germans have words such as "Tätergesellschaft" (perpetrator society) and "Vergangenheitsbewältigung" (coming to terms with [but also: conquering] the past), etc. In a nutshell, you could say that modern German identity has somehow a) embraced and b) rejected the Holocaust all at the same time - in the sense that some sort of "atonement" but also a "we are the exact opposite of that" has become a major theme!
You could arguably say that the culmination of this would have been the then German chancellor, Angela Merkel, giving a speech in the Knesset [Israeli parliament], declaring the security and existence of Israel to be an integral part of the German "Staatsräson" (roughly: purpose / reason why this state should exist / mission).
In this sense, anti-antisemitism can be viewed as a fundamental pillar of contemporary German national identity. But it is a partcicular kind of anti-antisemitism, that is, arguably, less grounded in personal conviction and more in an identity as "our grandparents or great-grandparents were perpetrators, and we atone for their sins". You could say that, in this sense, this is a very specifically Christian take on things (even though Germany is largely secular).
All good, until: enter immigrants! Apart from a huge number of people of Turkish origin, Germany is also home to the largest Palestinian diaspora community in Europe. As well as a variety of other Arab (and further majority Muslim) minorities. And many of them are refugees - either in the formal sense of "fled, obtained asylum" or in a more informal one of "emigrated by more conventional means, still getting away from something bad". Here is the problem: their grandparents and great-grandparents did NOT commit the Holocaust! The questions they ask when taught about it in "integration classes" tend to be more "could this happen to us, too?" and less "could we do this, too?". And by failing to adopt this identity of "Tätergesellschaft" - because, again, their ancestors did not run gas chambers - they are also failing to fully adopt a modern German identity. They therefore remain a "them", an "other". Because they lack the ultimate evil to atone for, they can never quite become the "but we are good now". The "redemption arch" that contemporary German identity is based on remains closed to them ...
... which, yeah, fine, we could all just file under "have fun, sociologists, sounds interesting and whatnot", if it were not for ...
a) Modern Germany having a far-right and a racism problem, and
b) the Israeli/Palestinian conflict
You see, Germany has a far-right problem on par with that of the rest of Europe (thing LePen, Orban, Meloni, ... you name it!). Their is called the AfD, and they, literally, include some bona-fide real actual neonazis (you know the "Hitler was a good guy, he just took things a little far" kind). They also have "even further than that" parties - as do we all - but those have mostly been muffled with the AfD becoming the 2nd strongest party in the country.
But because performative "anti-antisemitism" has become such a non-brainer in German identity even the "just about publicly defensible" part of the far right is of course also against anti-antisemitism. At least on paper. In practice, what they tend to really mean is "so long as them Jews stay in that Israel and - even better! - fight even browner foreigners, we can sort of pretend we like them". And, yes, I am deliberately using quite cynical language - because the whole thing is quite cynical!
So, therefore, you will find really far-right Germans - and, under the pressure they put on the mainstream: actually not at all far-right ones, such as the Greens, and the Social Democrats, caving in on topics that are just not that bad in terms of losing votes - now basically "outsourcing" the issue of anti-semitism to people who fail to successfully feel guilty for the sins of their forefathers - because, again: well, their gradparents really actually DID NOT do the Holocaust. And they hence find it hard to identify with the entire "we used to be the really bad guys - but we have atoned, and now we are really good" thing. The fact that - as I stated above - the whole narrative is super "Christian" and these people are by and large not, makes this even harder for them to grasp!
Hence: enter contemporary Germany! The country that committed the Shoa. Where even far-right natives are anti-antisemitism and even the left condemns how immigrants just do not commit to the "anti-antisemitic" identity the way ethnic Germans do ...
... and this ends up, ironically, really stinking of racism!
Muslims (says Habeck, in the video quoted) might - by default - be anti-semitic. Unless they specifically say otherwise! Arabs (said president Steinmeier today, demanding all Arabs explicitly distance themselves from anti-semitism and Hamas) might be - by default - anti-semitic, unless they explicitly say otherwise!
There is just one small problem: putting an entire group of people under suspicion of - for no other reason than their religion or ethnicity - being anti-semitic, whilst this obviously does not apply to yourself and people like yourself. Even worse when you happen to be projecting your own guilt onto people who ... have no reason to carry it ...
... that is actually really fucking racist!
In the meantime, the Deputy Minister President of Bavaria, the largest of the German states, is one Hubert Aiwanger - who has an openly anti-semitic, possibly even outright pro-Nazi (depends on where you draw the line) past. That caused a bit of a stir - but he still has the job! Nobody threatens him with deportation or withdrawal of citizenship. Because he is an ethnic German and ... well ... that is obviously altogether different than all those non-natives who ask the wrong questions at Holocaust education class.
... and that, long story short, is why I really do not think "but some German politician said" is a particularly good basis for anything antisemitism related!
For the record: I vehemently oppose antisemitism! I just also vehemently oppose it being instrumentalised in order to bash immigrants - and that is what is happening in Germany right now! It is very, very worrying!