The acronym YMCA stands for Young Men's Christian Association everywhere — including Mobile County, Alabama. There is no unique local meaning.
Mobile's YMCA dates to 1855 (chartered 1856) — one of the oldest in the South. It served as a hospital during Civil War yellow fever epidemics and hosted integrated programs during segregation (limited but notable).
Bottom line: YMCA = Young Men's Christian Association globally; Mobile's branch is a standard community wellness nonprofit.