Firstly, everyone is different wrt going for assessment. Some people will want to know, some people don't see a benefit to a diagnosis (although this can change in my experience)
For me, my need to know was greater. Probably part of my Autistic make up. So I went through assessment in my early 50s (ASC and ADD)
There is the benefit of meds, if required / applicable. This is obviously helpful when people are struggling constantly to quieten their minds and need to focus.
Secondly, and again everyone is different. Everyone will react to their diagnosis differently. They will use that information the way they want to (be it a comfort or reason to explain how they have felt their whole lives and how they move forward, or to access therapy and support)
It's also nice to think that you might unlock some understanding among friends and family. This is a tricky one, as not everyone understands.
Regarding 'identity' - again, a personal thing im exactly the same way as eveeyones symptom profile will be different. I have a lot of self awareness (I was told in assessment) so while I'm aware I may be struggling with things, or doing things that are deemed 'weird' I cannot physically stop myself at times. I find this embarrassing, so avoidance if often used. (As in I don't go out and don't mix) To an extent, my ND is part of my identity as it can consume me at times. Such is the nature of the beast (again, different symptom profiles present in different ways)
One thing I know is that I felt shit when I was undiagnosed. I didn't know what was real, and if I was struggling socially, I didn't feel able to say why. Upon diagnosis, I was able to tell a few people. I actually still feel a bit shit about that, as I feel like I'm now judged by some. So I feel sometimes like I can't win.
In actual fact, the amount of people now voicing that they think "diagnosis is given out like sweets" has made it all a whole lot harder.
Finally, regarding anyone who believes they have gotten to adulthood and already built effective coping strategies; please be mindful that this may not always work. I was managing just fine with my coping strategies, until I wasn't. Poor health and menopause around 49/50 years old meant I could no longer cope / mask / pretend. I had to seek help as I was having a breakdown by forcing myself to continue to cope.
It's debilitating, and not something anyone I know seeks diagnosis of due to it being novel or trendy.
Bottom line is, if you're struggling and it's affecting your life, then speak to the GP to get the ball rolling.
Anyone who thinks 'we are all a bit like that' well that's just fine if it doesn't significantly impact your life. Please though, don't make anyone who is formally diagnosed feel like a charlatan or minimise their struggles.