I'm guessing that's a 80286 AT clone. The first PC I ever built had the Award BIOS. I bought the kit from Jameco Electronics back in the late '80s and used it for many years. Wikipedia says Award was bought out by Phoenix in 1998.
I'm a bit surprised that the time-of-day clock overflowed. The standard for many years was set by UNIX, which stored the date as a 32-bit integer representing the number of seconds since January 1, 1970. 32 bit UNIX time doesn't run out in 2038, so obviously they something different, not wanting to waste a whole 32 bits back when memory was precious.
I found this note in the Wikipedia BIOS article: "Beginning with the IBM AT, PCs supported a hardware clock settable through BIOS. It had a century bit which allowed for manually changing the century when the year 2000 happened. Most BIOS revisions created in 1995 and nearly all BIOS revisions in 1997 supported the year 2000 by setting the century bit automatically when the clock rolled past midnight, December 31, 1999."