Southern "Mammy" & Irish "Mammy" - why the same?

[Exapno_Mapcase]

Online Etymology Dictionary:

mamma (n.)
1570s, representing the native form of the reduplication of *ma- that is nearly universal among the Indo-European languages (cf. Greek mamme “mother, grandmother,” Latin mamma, Persian mama, Russian and Lithuanian mama “mother,” German Muhme “mother’s sister,” French maman, Welsh mam “mother”). Probably a natural sound in baby-talk, perhaps imitative of sound made while sucking.

Its late appearance in English is curious, but Middle English had mome (mid-13c.) "an aunt; an old woman," also an affectionate term of address for an older woman. In educated usage, the stress is always on the last syllable. In terms of recorded usage of related words in English, mama is from 1707, mum is from 1823, mummy in this sense from 1839, mommy 1844, momma 1852, and mom 1867.

mammy (n.)
1520s, diminutive of mam (see mamma). Meaning “black woman having the care of white children” is by 1837, Southern U.S. dialect, variant of mamma.

No coincidence, and a long history that goes deep into the English language at far more points than two.