Put the following in B3 (credit to @Alexander-Ivanov for the countif condition):
={UNIQUE(A3:A),ARRAYFORMULA(COUNTIF(UNIQUE(A3:A),"=" & UNIQUE(A3:A)))}
Benefits: It only requires editing 1 cell, it includes the name filtered by uniqueness, and it is concise.
Downside: it runs the unique function 3x
To use the unique function only once, split it into 2 cells:
B3: =UNIQUE(A3:A)
C3: =ARRAYFORMULA(COUNTIF(B3:B,"=" & B3:B))