If there were a Second Sun we would notice a large number of changes due to:
1. increased mass in the inner solar system
2. increased energy emitted by the pair of Suns
3. greater angular diameter of lighting
The second Sun would immediately begin to change the orbits of the planets. Star charts would clearly be wrong. The planet Mercury would certainly be
moved from its orbit in quick fashion. The tides would change since the Sun is about 1/3 of the tidal effect and a second Sun would increase the solar
component of the tides.
The increased energy reaching the Earth would spell doom. A 10% increase in energy reaching the Earth would cook all of us.
Sunrise and sunset times would be altered by the new light source that is further out from the Sun.
During daylight hours we'd experience shadows going in more than 1 direction. This can be seen at night walking through a parking lot lit by multiple
These fairly showy effects would quickly tell us all that a second Sun is no fun at all.