If you discard a character to power them up does it affect all characters you control with that name?

For example, if you have multiple Army version of the same character out, or multiple of unique characters due to Scarlet Spider, duel Nature, Dr. Light etc and discard that same character do all of them get the +1/+1 or just a "target"?

No. Powerups only applies to a particular character on the field for each one.

And do not forget that the character must be in combat.

- error, sorry -

The rules state that a power-up is a targeted effect.

As such, it only affects the targeted character.