Oh, yeah, my password manager is full of answers to 'security questions' like 'place of birth', 'name of first pet' and 'high school name'. Just about the only time that gets awkward, is when the same answers turn out to be required during certain phone conversations. Yes, I do in fact, happen to be born in Qjhdfoudvprx -- quaint little, very traditional village...
I used to give random strings like that. Next time you get asked one on a call, say "Oh I hate those intrusive questions so I just mashed the keyboard." Many support agents will accept that as a valid answer!
Yes, well, unless the poor CSR cannot actually see your answer, but has to enter it for a 'yes/no' response. Had a fun time with Rackspace support around the turn of the millennium, where I had used the output of 'openssl rand -base64 16' or somesuch, and that had to be entered verbatim, with the person on the other side apparently not understanding what a 'forward slash' was, and, yes, it was urgent...
Anyway, I don't find these questions intrusive as much as 'not fit for purpose', with a bit of (cultural) insensitivity mixed in. I once had to console a very-upset very-serious person from an African country with regards to the the 'name of pet' question they just had to answer as part of a mandatory signup for a US-based third-party service, and which they just couldn't find an answer to.
And whenever I'm asked about "name of high school" or (even worse) "high school mascot", I'm tempted to flip out with something like "yeah, which of the seven-or-so I attended, for no-fun-at-all reasons, and most of which I don't even remember, but I can assure you none of them had a mascot", but then I just take a chill pill and mash my keyboard...
I do the same, but I have been using random words.
So for my gas bill, my pet name is `chlorine wastrel percept`.
Easier than pure randomness, and strong enough for the purpose.