(previous comment removed) I'd like to show a step proof of the 2nd DSolve result or say nothing - I do not have it at the moment.

I had made a careless error in my head while was assuming mm was right. I have no proof to show you at this time why the internals of Mathematica made it's decision and would rather say nothing as I said before.

But since you re-asked, I will say this is not an infrequent question on Communities and that often questioners have set up conditions incorrectly. In this case I will say "valid conditions do not always help Mathematica solve a PDE". The code used to solve is on your system (you can find it through Help). But that is not an answer because I have not read "all of how mm solves each PDE" I cannot answer you directly, as to internals. I wish to avoid taking an unresolved question and adding further guesses.

The Help documentation mentions choosing c1 after solving (not during), I'm unsure if there's a reason why. This may hint that, in effect, one should select a function rather than a constant. However, this is guess work.

DSolveValue[{D[f[x, y], x] + D[f[x, y], y] == 2}, f[x, y], {x, y}] == 2 x + C[1][-x + y]
DSolveValue[{D[f[x, y], x] - D[f[x, y], y] == 0}, f[x, y], {x, y}] == C[1][x + y]
DSolveValue[{D[f[x, y], x] + D[f[x, y], y] == 2, f[x, 0] == x}, 
 f[x, y], {x, y}] == x+y

But that is a poor answer, since combining these still does not solve, even though Mathematica "knows how to solve systems of PDE", and because it is a guess not a fact, and because the above is not your original system. The problem is then to know how the internals work exactly, and I have not memorized all of the internals.

DSolveValue[{D[f[x, y], x] - D[f[x, y], y] == 0, 
  D[f[x, y], x] + D[f[x, y], y] == 2, f[x, 0] == x, f[0, y] == y}, 
 f[x, y], {x, y}] == Identity