StreamPlot No Longer Works With Piecewise Function?

It does seem to be a bug. As a temporary workaround, you can re-express your function in terms of either UnitStep[] or Boole[], like so:

f[x_] = Dot @@ MapAt[Boole, Internal`FromPiecewise[
        Piecewise[{{2 x - 3, x > 1}, {-x, -1 <= x <= 1}, {2 x + 3, x < -1}}]], 1]

StreamPlot[{y - f[x], -x}, {x, -5, 5}, {y, -5, 5}]

Note the use of the undocumented function Internal`FromPiecewise[] for decomposing the Piecewise[] object.

A workaround:

g[x_] := -Boole[Abs[x] < 1] x + Boole[Abs[x] > 1] (2 x - 3 Sign[x])

Now,

StreamPlot[{y - g[x], -x}, {x, -5, 5}, {y, -5, 5}]

yields an incorrect plot!

Using Graphics:

Graphics[Catenate@
  Table[{Arrowheads[0.01], 
    Arrow[{{i, j}, {i, j} + 0.2 Normalize[{j - g[i], -i}]}]}, {i, -5, 
    5, 0.4}, {j, -5, 5, 0.4}]]