CRJ-200 flight crew reported a Ground Conflict with another aircraft during taxi out for take-off.

Narrative

We had just finished deicing at the central de-ice facility in ZZZ. It was actually snowing decently; taxi surfaces were lightly contaminated. The time was approximately XA20Z on Day 0. We had deiced at bay X; Aircraft Y has just finished in bay Y. Deicer switched us over to Pad Control; and the First Officer called for taxi. The instant they finished the transmission; before the Controller could respond; Aircraft Y stepped on us calling for their own taxi clearance. The Controller somewhat audibly rolled their eyes and gave us our clearance. I started moving as my First Officer (FO) read it back. I was moving south on the taxiway as Aircraft Y read back their identical clearance. I taxi past bay Z and as I'm approaching bay Y I see Aircraft Y moving kind of fast toward me; moving up from Y to Z; but at this point my nose is directly in front of bay Y's exit. I think to myself; 'wow this plane is moving kind of fast considering they got to stop in like 50 ft.... I'm CLEARLY right in front of them.' Then out loud I say 'Surely they see me; right? [Wow!] This person wants to actually hit me.' And I hit the brakes; firmly; to prevent that. I thought I had managed to do it smoothly enough that no one would take much notice; but after the flight the Flight Attendant (FA) asked me; 'so what happened as we were leaving deice?' And said they definitely noticed the sudden stop. So; Aircraft Y speeds past; cutting the corner in front of me; wing tips missing by probably less than 20 ft. I turn on every light and key the mic with an 'Aircraft Y; wake up!' as I see the First Officer looking backwards out their window at me; the plane they almost became a part of. We told the Controller we just got cut off; ATC said sorry that was a 'coordination mistake releasing two aircraft to taxi' Okay; but I was a lot more concerned with the airline Captain who was either blind and didn't see a plane in front of them; or was so committed to forcing their way in front of me that they were willing to cause a collision over it. Another aircraft (probably the one sitting in another bay that Aircraft Y would have had to squeeze closer to as they cut the corner in front of me) chimed in saying; 'Aircraft Y is going WAY too fast.' Aircraft Y switched to Ground's frequency without another word; but continued to taxi quite quickly despite the contaminated taxiways. Ground told us to follow them down Kilo; which we did from a safe distance; partly by choice and partly because they were impossible to keep up with. Aircraft Y's fast; aggressive taxi also caused them to turn the wrong way on K3. The Ramp Controller should have specified for Aircraft Y to give way to us; who were clearly ahead of them; and had been given our taxi clearance like 20 seconds prior to them. But that aside; even if the Captain of Aircraft Y thought that they were supposed to go first and that I was the one somehow cutting them off; it takes a special kind of hazardous attitude to see an aircraft perpendicularly right in front of you and be like; 'No; I'm getting in front at all costs.' I have to give way to surprise airplanes at least once a week at ZZZ; but you drive defensively and if you never assume you have the right of way. The kind of unwritten rule of ZZZ taxi is: don't hit anything and don't stop taxiing unless it's to avoid hitting something.