I don't get, what's the meaning of 'remember your bitcoin password' do BTC need password to operate or its talking about bitcoin wallet? Even if it's about the wallet well that sounds like a centralised wallet that ask for email and Password to log in.
Actually, there's no such as a bitcoin password.
You're right, that's describing about the bitcoin wallet's password and likely in general, all that requires login details. Just do not forget about that which also includes the exchanges since you might have deposited your bitcoin there.
My thought is that, the reason for this rule is for privacy concerns, but then, I might be wrong, so my question is "are there or is there any security issue(s) related to using or reusing a particular or one single bitcoin wallet address for a very long period of time?
It's probably because someone can trace you and can use that against you in a scenario that we don't know might exist in the future. It's lessening the privacy you have upon using it again.