One more theoretical, one more practical.
Theoretical first. The way a 3 way switch setup works is that you have a couple of wires which run from one 3 way switch to the other; they connect to the 2 “traveler” terminals of each switch. Separately, there’s a wire from the power source which connects to the “common” terminal of one switch, and a wire to the fixture which connects to the common terminal of the other switch. What I’m wondering about is the color of the two traveler wires.
I have some 3 way switch setups in my house, and they have a black, red, and white wire in the connecting wire. So you have one black wire from the power (or fixture), one black wire as a traveler, and a red wire as a traveler. ISTM that this presents opportunities for confusion, since if you remove and reinstall the switch you’re looking at two black wires, one of which needs to be connected to the common and the other to the traveler. It would make more sense for the connecting wire to have two reds and a white, so that all reds connect to the travelers and all blacks to the common.
What’s the rationale for doing it the way it’s done? (Is it possible that these wires are also used for other purposes where the tri-color scheme is more appropriate?)
Second, the practical. I’m thinking of adding an outlet to a room, midway between two other outlets. The wiring for additional outlets seems pretty straightforward, but that’s assuming that the outlet you’re connecting to doesn’t already have another outlet feeding off of it. In this case, I would have to think both of the outlets on either side are hooked up to each other, such that there will be no free terminals on the outlet I want to use. So the question is: is there any reason I can’t hook up two wires to the same outlet terminal?
(The other option would be to (hopefully 
) find the existing connecting wire at the point where I want the new outlet, cut it and splice in some new wires and hook those up to the new outlet. But this is an exterior wall and the existing wire would be behind insulation, and this fishing and splicing is complicated enough. As it is, I’m thinking of running the wire down the wall and along the floor behind the floor trim and then back up into the wall.)
The reason that there aren’t two reds is: Romex cable doesn’t come that way. You can get W/B or W/B/R, but not B/R/R or W/R/R.
As for your outlet - it depends on the age of the outlet. If it’s old, and requires the wire to be wrapped around the screw, then you can’t do it. If it’s newer, and allows the wire to be inserted between the square washer and the bottom plate, then yes, you can attach another wire to that terminal. Or, you can attach a short pigtail and use a wire nut to connect the two wires.
Also, you can use the backstab connectors in addition to the screw terminals, but I don’t recommend using them, although they have improved a lot over the years.
The reason that there aren’t two reds is: Romex cable doesn’t come that way. You can get W/B or W/B/R, but not B/R/R or W/R/R.
But the question then is why doesn’t (3 wire) Romex cable come that way? Unless it’s more commonly used for other situations where the 3 color setup works better.
If it’s newer, and allows the wire to be inserted between the square washer and the bottom plate, then yes, you can attach another wire to that terminal.
It’s not newer, but it’s not a big deal to just replace that outlet with a newer one. They don’t cost much.
Demand. 3-way light switches are a lesser demanded usage than say 220volt systems using red and black as 2 legs and a common neutral. Also one can use a switched receptacle on red for instance and an always hot outlet on black. I have power in my dining area running from the switch to the light on red as well as an always hot black lead feeding the pull cord for the ceiling fan. Bathroom exhaust fan and light are the same way except the fan is switched too.
You can’t get red blue and white either but in commercial buildings fed by 3phase with a 208 volt panel, the 3rd leg beyond red and black is many times signified by blue.
In regards to your outlet, if room exists in the receptacle box I’d pigtail it even with a new outlet. I don’t like running loads through a device if I don’t have to. And if you are replacing an outlet, if it doesn’t exists already, I’d put a GFCI one in at the furthest upstream location, even if not required by code.
Having two red wires in a Romex would cause problems if you care about which wire is which. You don’t for travelers in a three-way switch, but you do if you’re wiring a fan where one wire is the light and one is a fan, or an outlet where one wire is switched and one is constant on. Of course you could have a red and a blue wire that would solve both problems, but using a red and a white for travelers isn’t seen as a problem. Usually it’s pretty obvious because the cable with only a black is the source or the fixture wire, and the cable with a red and black are the travelers.
My own house they used “Greenfield” where you pull the wires into a conduit afterwords, so you can pick the color wires. They tended to use two brown wires for the travelers.