It is quite independent. They are in a tug of war with the commission and the council of ministers by design, so their autonomy fluctuates with time, but the commissioners (and several member-states) are regularly quite unhappy with the ECB. The ECB typically does not care about politics and looks only after economical indices. They are known to be afraid of the tiniest bit of inflation, which resulted in counter-productive policies at time.

I am not familiar enough with the federal reserve to say how similar they are.