The connection between TOS and the late 22nd century was established in the second pilot, "Where No Man Has Gone Before". Tarbolde wrote a poem in 1996, and Gary Mitchell said something about it being one of the most passionate sonnets written in the past couple of centuries.
The Wrath of Khan was two-faced on the dating - the title card identified the time period as the 23rd century, yet Khan said that he ruled a large part of Earth two centuries before, which would place his rule in the late 21st century.