section, bureau, dept, class, copy, part, portion, counter for copies of a newspaper or magazine
部 centers on dividing a whole into smaller organized units: a section, department, or class, and by extension a part or portion of something. The counter sense for copies of a publication fits the idea of a numbered part of a series.
department (in an organization, company, etc.); division; bureau; faculty
all; entire; whole; altogether
portion; section; part
room; chamber
one part; one portion; one section
club; fraternity; sorority; clubhouse
interior; inside; internal
force; unit; corps; squad; troops; party
headquarters; head office; main office
outside (e.g. of a building); exterior
division (of a larger group); branch; field; class (subclass); group; category; department
subordinate person
head (chief, director) of a section or department
parts; accessories; components
northern part; the north (of a region)
management; (executive) staff; leaders; leadership; top brass; upper echelons
southern part; the south (of a region)
faculty (of a university); school; college; department
branch; branch office; subdivision
club activities; extracurricular activities
top part; surface
part (esp. of the body); region; site; cut (of meat)
one's post; one's station; department; bureau; section
lower part; substructure
部 combines 咅 (a phonetic element carrying the sound) with ⻏ (阝, a variant of 邑 meaning a place or settlement), suggesting an administrative division or section of a settlement.
The left side 咅 looks like a standing person 立 with a mouth 口, and the right side ⻏ is a place. Imagine a person speaking to organize people into sections at a bureau.
For ブ, picture a bureau with many sections, and the sound of the word 'boo' echoing through the halls as someone calls out section numbers: boo -> ブ.