So my answer really comes from waaay back when I was a producer.

What is the psychological effect of listening to two different pieces of audio simultaneously?

It depends on the audio content, its message if it has any, and the psychological state of the listener. It's generally not dangerous to try it out, but most often it will sound like incoherent garbage.

What if the pieces are the same and start at different times?

Now that is a more interesting question!

Up to ~15 milliseconds — You won't perceive the two pieces as two different pieces. It is equivalent to running through a comb filter, which gives (in my opinion) a weird "woody but also digital" sound. It's like a flanger effect that "doesn't move". Basically, it'll sound like an "off" version of the original piece. Comb filters are more often "used" by accident because there are two versions of a performance with delay times (for example, when two microphones record the same instrument).

~20 to ~50 milliseconds — Starts to sound like multiple performers playing the same piece, or like a "thicker" version of the original piece.

~40 to ~120 milliseconds — "Slapback" delay. It's starting to sound like multiple parts. Also makes parts sound "thicker", but very distinct to 50s music, blues, and rockabilly.

~150 milliseconds to several seconds — Starts to sound like an echo'ed version of the original piece.

~More than several seconds — One of two things could happen:

1. The original piece has a rhythm, and the delay "syncs up" with it. Then, you get a new piece that hopefully sounds cool. If the delayed version is the same volume as the original version, the overall piece will get louder when the delayed version starts.

2. Any other condition: It sounds (musically) incoherent to the listener, and will basically be perceived by the listener

All of these delay effects are musically useful, but also serve as the prototypes for other effects. Additionally, digital equalizers are mathematically equivalent to a bunch of delays "stapled together" and rescaled.