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