Tags don't bother me at all; I've also heard "Highly Sensitive Persons" have a high sensitivity to tags.
I understand your concerns about LARPing for accomodations, but the reality is people do legitimately have traits (like not being able to focus when people are talking) that aren't a "disability" or "diagnosis". I very explicitly picked my example because there's an autistic comedian who was on the verge of being fired at work for her autistic social skills but she failed her first examination because they asked her, "Do the tags on your shirt bother you?" and she said, "No." And the psychologist interpreted that to mean that she didn't have sensory issues, and therefore she didn't have autism, and therefore she could be fired for being socially awkward.
But, as she explains, of course the tags on her clothes don't bother her because she has a system! And so she had another exam and was more forthcoming about what they were looking for and now she DOES have autism and she CAN'T be fired for social awkwardness.
You can have a bona fide issue with your work environment, but it has absolutely zero value unless it's part of a recognized disability, even if it can be independently verified.
The real problem is that employees (in the US, anyway), can be treated with absolutely arbitrary indifference, including being fired for absolutely no reason.
In that environment, the ADA and the Civil Rights Act ^(and unions, which are in superscript for a reason)^ are the only legal paths to be treated with respect and dignity