The interpretation is not correct.
The $D$ statistic is the largest observed difference in ecdf, and as such is a way to describe the magnitude of the difference in distribution (one of many possible ways). Which is to say, your colleague's description of $D$ is more or less correct.
However a p-value is not the likelihood (nor even the probability) of the observed measure of difference.
(In the present context) It's the probability of observing a $D$ statistic at least as large as yours, when the two samples were drawn from the same continuous population.
Those two things (what you report your colleague as saying and what I wrote) are quite distinct notions.
May find it useful to read this (and the actual ASA Statement on Statistical Significance and P-Values itself, which immediately follows; each is only a couple of pages long):
Ronald L. Wasserstein & Nicole A. Lazar (2016),
The ASA Statement on p-Values: Context, Process, and Purpose,
The American Statistician, 70:2, 129-133,
It does explain what a p-value is and what it isn't, though it doesn't address all the ways in which people misunderstand p-values. It also doesn't list all the criticisms of p-values nor everything people may do to supplement or replace them. (This is not intended as a criticism, just a description of some of what it contains.)
You might also get some value from
Ronald L. Wasserstein, Allen L. Schirm & Nicole A. Lazar (2019),
Moving to a World Beyond “p < 0.05”,
The American Statistician, 73:sup1, 1-19,
if somewhat less directly relevant to the present question. (I don't necessarily agree with everything in this article but that's not necessarily a problem.)
"Likelihood" in ordinary conversation is more or less interchangeable with probability, but in statistics they're quite distinct concepts.
There were three issues with what you said in relation to the p-value; one was using the word likelihood when you mean probability; the second was the probability you gave related to the wrong event (which you've fixed in your restatement); and the third was it's actually the probability of that event given that the null hypothesis is true (i.e. given that the populations that the samples are drawn from are the same) -- and given the usual assumptions of the test, naturally.