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 (Topology and its Applications 158 (2011) 2535-2555)

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

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 (Topology and its Applications 158 (2011) 2535-2555)

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