How come FFT window give half size frequency bands when filter can seems to have an infinite precision?
The DFT is a tool to represent the spectrum of any signal. Usually for an N point DFT only N/2 output coefficients are required because for a real signal the DFT is symmetric about the centre.

No filter has infinite precision, neither infinite bandwidth nor numeric representation. A filter with enough precision can be chosen for an application to seem smooth enough.

The DFT though doesn't have anything to do with a filter. It simply represents the spectrum of the signal at it's input.