Not normally, that wouldn’t make much sense. It’s easier to just say you signed an NDA and anyone with sense knows you’re not allowed to talk about the work.
Biologically and functionally speaking, he’s whatever and everything. Practically speaking, he chooses to have a dick and look like a big muscular man that looks nothing like his previous form (so its even more deliberate), so it seems the only thing that survived his ascention was his gender identity.
In my headcanon, the further into the future you go the less attached to his humanesque form Dr Manhattan becomes, and eventually he just becomes a sentient force that does things that are beyond comprehension without needing a physical body at all.
Even if you don’t mention who you signed it with? There’s quite a bit difference between saying “I’ve signed NDA” and “I’ve signed NDA with company (name)”
Breaking the agreement does not always mean you will get in trouble for it. This is the one rule that can be risked breaking without facing repercussions.
You’re saying you got out of college ten years ago, and worked for exactly one company that entire time, and you can’t say who it was or what you did because of an NDA?