It happens but again it's not at all universal. Scott Alexander seems to think emotional blunting is a legitimate effect of SSRIs, not just a correlation–causation confusion. He also notes that

There is a subgroup of depressed patients whose depression takes the form of not being able to feel anything at all, and I worry this effect would exacerbate their problem, but I have never heard this from anyone and SSRIs do not seem less effective in that subgroup, so these might be two different things that only sound alike.