What is the term for multiple simultaneous vocal lines?

It's certainly a polyphonic device, and 'polyphony' is another name for 'counterpoint'.

1. What is it called when people sing different lines at the same time?
2. What is the term for music with two or more vocal lines?

What is it called when people sing different lines at the same time?

When two singers sing different lines at the same time it is called "rounds".

What is the term for music with two or more vocal lines?

Polyphony, in music, the simultaneous combination of two or more tones or melodic lines (the term derives from the Greek word for “many sounds”).