I think a workable definition for "depth" would be one based on the number of "skill levels" the game has, in tournament play. As I recall from the last time this subject was hashed around here, Chess has something like, say, 15 ratings levels. If my rating level is one higher than yours, then I have a 60% chance of winning a game with you (on average). I believe Go has many more ratings levels -- say, 30. In this sense, Go is objectively much deeper than Chess. In the same sense, the depth of Chess and WtP could theoretically be objectively compared -- but only if WtP had tens of thousands of rated tournament players (which, of course, it never will). Still, we can speculate on this basis...

(please note: I'm not certain of the particular numbers, but I do recall that Go has significantly more rating levels than Chess).