An ID (I’m thinking specifically of a drivers licence in Australia) usually contains virtually all information anyone would ever need to identify themselves as you over the phone - name, address, date of birth, drivers licence number, etc. Having someone scan this and save it is not something you really want anyone to ever do.
I'd assume the information on the 2d barcode is the same as what's printed on the front. The problem is that you have no idea whether a scanner is connected to a larger surveillance database. A GDPR-style law and real enforcement would go a long way to restoring some societal trust in the US. As things stand right now, it's prudent to put tape over the 2d barcode and the printed ID number, and only remove it when necessary (eg actually interacting with the police).
Well, your address is printed right on the front too. I thought eye color was printed on the front, but I guess not. So perhaps there are a few fields like that.
I wouldn't think printing out your own 2d barcode with some fields masked would be strictly illegal, as long as you weren't committing fraud by putting fake info.
Practically though, even just covering mine up with tape I've run into a few people who call it a "tampered" ID. IIRC this has only happened for situations where I accept they're going to require the identification number, so I've removed the tape from the printed number on the front but not the barcode on the back (because it's harder to nicely put the bigger piece of tape back). When I hand over my ID I ask them to key in the number instead of scanning, and most people will happily do so. But the occasional person will get tense and say it's a "tampered" ID and refuse to accept it. I've always quieted them right down by removing the back tape and handing it back to them. But I can imagine if you have a different barcode and someone notices it might set off similar "serious business" flags that you won't be able to assuage so easily.
But if you've got the bandwidth to try it, go ahead. Human rights aren't going to defend themselves! Just remember to be pleasant in your interactions, and it helps to have examples ready of why you want to keep your information out of databases.