While browsing this answer I noticed the sheepskin cover and recalled that I couldn't remember a cockpit that didn't have them. It's not entirely surprising that they're used, but I am surprised that they seem to be always used. Are they indeed ubiquitous? Are there any competing alternatives? Do modern airliners use them as much as GA/military/sport aircraft?
They aren't "almost uniformly sheepskin", but they are absolutely uniformly flame-resistant (or self-extinguishing) as required for aircraft certification.
Sheepskin (and other natural wool fabrics) are both flame resistant and self-extinguishing (which is why industrial fire blankets are often made of wool), and sheepskin tends to wear well, so it's a good choice for the two seats that will absolutely always be occupied when an aircraft is being operated.
In terms of comfort, sheepskin covers also breathe well (so the pilots aren't sweating profusely while operating the aircraft - one of the reasons they're popular accessories for cars).
Other popular fabric choices are natural leather, flame-resistant synthetic leather (vinyl), and various other textiles which pass the FAA "burn test" requirements (test procedures are described in Part 23 Appendix F for light aircraft, Part 25 appendix F for transport category aircraft).