Given how similar they are, evil warlord bit notwithstanding, and how they both have an infinite number of divergent "selves" floating about?
A story about the two would likely be a catty argument with both of them insisting that the other is an offshoot variant, preferably an argument hurled through cosmic loudspeakers as their respective vehicles hurdle wildly through the membrane of time and space.
Eventually, Kang's future self Immortus would step in, be a total dick, and the two of them would begrudgingly cooperate to upset his plans. The Doctor would take off at this point, making a casual remark that's actually super devastating and makes Kang pretend to be magnaminous and "permit the Doctor and his silly box" to leave, to cover how afraid he is.