Which, of course, leaves the question why on Earth the state and the religions should use the same word for it, why the legal and the religious aspects should go under the same term. I can see absolutely no necessity for this; it's a relic from times where church and state were far closer intertwined than they are now.
It would be far better in terms of separation of church and state if legal contracts that don't involve religion never involve the term "marriage" at all - especially now that a precedent has been set in calling (some of) them civil unions -, and if the religious rite called marriage, in turn, will have no impact at all giving anyone involved any legal rights and benefits whatsoever.
We absolutely are able to take the state out of the marriage business, and in terms of secularity, that's exactly what I think we should do. (In the US, that would be even easier to do than over here... I generally think the German constitution is a wonderful document that I very strongly support, but I envy Americans over one fact: unlike ours, your constitution does not mention a "special protection of marrriage by the state"; it doesn't mention marriage at all). That doesn't change that responsibilities, insurance, protection of children etc.pp. can (and likely, must) be legally regulated in some way - but that's what the civil unions would be for, you don't need marriage for that.
And of course, no religious officiant should ever be able to preside over the civil union ceremony (only judges, magistrates etc. should be able to do that), just as no judge, magistrate etc. should ever be able to perform the rites of marriage (only priests, rabbis, etc. should be able to do that).
BTW, I wholeheartedly agree with what you said at the end, except for the use of the term "marriage": The state should not limit who gets access to a civil union. No matter the number, sex/gender, orientation, race/ethnicity, religion, type of relationship, etc. of the participants - everyone who wants to get such a legal union for their household should be able to get them. Fullstop. Anything else would be unfair discrimination, and a needless and inacceptable violation of equal rights.
Religions, however, should forever remain allowed to limit access to the privilege of marriage rites in whatever way they see fit, with the state forbidden to meddle in that decision. That's simply freedom at work.