Bug in DSolve when solving a system of two second order PDEs?

I’m solving a simple system of PDEs in Mathematica 11.1:

DSolve[{D[f[x, y], x, y] == 0, D[g[x, y], x, y] == 0}, {f[x, y], g[x, y]}, {x, y}]

This gives:

{{f[x, y] -> C[1][x] + C[2][y], g[x, y] -> C[1][x] + C[2][y]}}

I expect the general solution to be:

{{f[x, y] -> C[1][x] + C[2][y], g[x, y] -> C[3][x] + C[4][y]}}

What do you think?

Thanks!