Sunday school's own history is of an English literacy program that narrowed into religious instruction once state schools took over general education. A comparable narrowing happened independently inside Jewish communities, without either tradition borrowing the idea from the other.
Talmud Torah schools, communally funded institutions teaching Hebrew reading and religious texts to children who could not afford private tutoring, existed in Jewish communities for centuries before Robert Raikes opened his first classroom in Gloucester, serving a purpose closer to general as well as religious education in societies where Jewish children were often excluded from, or segregated within, non Jewish schooling. As state education systems expanded across Europe and North America and Jewish children increasingly attended those schools for general subjects during the week, supplementary Hebrew schools, often meeting on Sunday mornings or on weekday afternoons after regular school, took over specifically religious and Hebrew language instruction, the same functional split that happened to the English Sunday school once government schooling absorbed literacy teaching.
The result in both traditions is a children's institution named for when it meets, a Sunday morning or an after school afternoon, rather than for what it teaches, because what it teaches changed once a different institution took over the half of the original mission that could be handled elsewhere. Neither tradition needed to copy the other to reach a similar shape from a similar pressure.