The glyph origins of 半
Perhaps those etymologies are not quite correct.
The, current, definition in the new Outliers dictionary gives the following explanation:

FORM
Ancient form:

半 was originally composed of 八 “to divide; cut into two parts” and 斗 “measuring spoon,” indicating the original meaning “a spoon used to serve alcohol with a capacity of half a 斗 (dǒu).”

They give the following reference:

杜忠誥，2002《說文篆文訛形釋例》，台北市：文史哲出版社，2009年2月初版修訂。
p. 191-195

Then under Meanings they continue:

1 (orig.) a spoon used to serve alcohol with a capacity of half a 斗 (dǒu)
2 → half

I guess a 半 is almost like a half-pint - if we’re to put it in modern terms westerners could understand.
——
edit: I would imagine 伴 is comprised of 亻/人 [a meaning component] and 半 [a sound component].