Asking about such things excessively is being nosy.
Illicitly acquiring the answers to these questions (e.g. by rifling through other employee's private information without permission) is instead a violation of privacy.
Nosy is a derogatory term which, in a business context, may make it sound like you, rather than the employee, is at fault. From the viewpoint of a 'nosy' person, the other party is 'hiding something', just like how calling an employee a 'crybaby' may result in others wondering if in fact the issue is that you are 'insensitive'. Derogatory character trait terms like these tend to come in pairs. If you do feel the need to include a character-trait based derogatory term for this, "Intrusive" sounds better than "nosy" in formal contexts. You should really avoid both, though.
A violation of privacy is an event/action and taking grievance because of it makes your position sound neutral and supported. From the perspective of the privacy-violator, others will assume, your privacy didn't matter, and that claim is socially unacceptable. (It also provides actionable information the people you have a problem with can use to change their behavior in the future, but that's better left to workplace.se to discuss)