This is/was true only for selected components like the JT3-Ds (KC-135E) and the taller tail cap.
It is a common myth that the KC-135 is a modified B707. In fact, the reverse is more true. The KC-135 design was actually finalized before the B707's. The 707 profited significantly from the KC-135 design.
The KC-135 has a forward crew access hatch, smaller fuselage diameter and laminated wood decking in addition to accomodations for the fuel bladders. Anyone who has laced those in will appreciate the fuselage diameter difference. The wings, wingroots and flaps have significant geometric differences too between the 135 and 707.
Some commercial 707s have been modified as tankers for the some clients and some commercial operators.
While it was true the KC-135E upgrades used the engines and the vertical tail salvaged from scrapped 707 airliners, I believe some other 707 structural parts were used on the KC-135E's. But the KC-135R upgrade was much more serious, because it required wing upgrades to accommodate the four CFM56 engines.