Replacing a electric receptacle in the middle of a series

One weird thing to my eyes is that white and red are on the same side.

Yeah, that's broke-a-doodle. Either someone wildly miswired other points in the circuit, or this tripped the breaker instantly when someone threw the switch, or it "burned out" one of the backstab connections which happens all the time.

Another problem is determining which wire in each pair is from upstream or downstream.

Doesn't matter. Work it out; regardless of what the answer is, the wiring will be identical. So it is not a thing you need to know.

Am I supposed to pigtail the pairs, along with a third wire that goes to this outlet? Mentally that doesn't seem like the right thing, but I couldn't really tell you why.

It literally does not matter. You can pigtail, or you can use a receptacle which allows an onboard splice. Your call, you can do it any way which suits you.

For instance, I usually pigtail because I'm attaching quad outlets on metal boxes in stress positions like on ceilings or under desks. Much easier to assemble the domed covers and pigtails on an ergonomic bench!

The only exception is certain applications where pigtailing is required. Safety ground, obviously. Or neutral in certain types of circuits.

What would be the "correct" replacement receptacle for this usage?

Not the 50 cent cheapie in your photo. If you graduate to the $2.50 spec-grade outlets, most of them have a feature called "screw-and-clamp" where each screw also functions as a plate clamp, that can grab up to 2 wires. Easy peasy.