Your number $b(U)$ is usually called the "character" of the ultrafilter $U$.  

In general, there may be uniform ultrafilters on the same set with different characters.     For example, it is consistent with $2^{\aleph_0}=\aleph_2 $ that some ultrafilters have character $\aleph_1$, others $\aleph_2$.   

Also more complicated "character spectra" are possible, according to [Sh:915](http://shelah.logic.at/papers/915) (Topology and its Applications 158 (2011) 2535-2555)

(Of course, if $2^\kappa=\kappa^+$, then all uniform ultrafilters have
character $\kappa^+$.)