While they are related, very closely, they are not the same airframe. The KC is skinnier and IIRC, shorter. The AWACS (E-3), the JSTARS (E-8) and the Hermes/TACAMO (E-6) do use the 707 airframe. Some countries, Israel for example, have tankers based on the 707 airframe, often conversions from old airliners. But the USAF KC-135, all the EC-135s and RC-135s (and at one time a few plain C-135s) are a distinct aircraft from the 707. The 707 first flew a few weeks before the contract for design of the -135 was let.
Thanks for the detailed info. I guess it would be more accurate to say the KC 135 is from the same airframe as the 707.
But, I assume the vintage is the same? It sounds like you would know better than I.
By “from the same airframe” I mean it to be BASED on that airframe.