I’d always understood that an owner of Aldi’s owned TJ’s, but Aldi itself did not. (According to Wikipedia: “From 1979, it was owned by Aldi Nord’s German owner Theo Albrecht until his death in 2010, when ownership passed to his heirs.”) But there sure seems to be a substantial amount of overlap in products. Anyway, looks like we (in the US) are getting Lidl soon.
EDIT: What @FFabian said.