They can have a mortgage on it. When they ask if it's Primary, Vacation, or Investment it due to the terms of the loan and flexability.
A primary Home can get up to 100% financing, or qualify for special programs (FHA). A vacation home is seen as a higher risk and will be hard to finance 100%, knowing if you have a mortgage on your primary you will pay that before your vacation also most will require 25% or more down. Investment properties can be a little tricky, if it's for a rental some banks will allow you to add 6 months of the rental income to your Gross helping you to qualify for the loan, but on the other hand most will require 25-30% down paymenat.