According to the manual, my receivers default setting is 6Ohms (There is an option to set from 4-6Ohms).
So I was thinking, would this receiver be happiest with speakers that are likewise 6Ohms? Does it really matter? According to the specs, it looks like I would get a little more power as well going with 6Ohms.
No, your receiver will be happiest with speakers that are easy to drive, meaning phase and impedance remain somewhat constant and high-ish (e.g. 8 ohms).
The 6 ohm setting means the speaker's internals are configured to still be okay with speakers that dip a little lower, so you can hook up those kinds of speakers. The 4 ohm setting means the same thing.
You get a little more power into 6 ohm speakers versus 8 ohm speakers because of how things work, but you might not get better quality sound to match, so you should consider the power increase an incidental rather than something upon which to base your choice of speaker.
If you were to hook up speakers that go to 2 ohms, then your receiver would be very unhappy.