The German Albrecht family that owns Aldis, among others, purchased TJs back in 1979 from the founder of TJs, Joe Coulombe. The German family patriarch died in 2014 and his heirs inherited the company lineup, including TJs.
You're kind of going in the right direction, but the facts you stated are off quite a bit.
Aldi's origins go back to the early 1900s when a certain Mrs. Albrecht founded a small local store in Essen, Germany. Following WW II, her sons, Karl and Theo took over the store, changed the merchandising concept and expanded throughout Germany and other parts of Europe under the name "Albrecht Discount", which was later changed to Aldi.
In the early 1960s, after a disagreement on whether the store should, or should not, sell cigarettes, the business was split into 2 entities known as Aldi Nord (North) and Aldi Sud (South). Karl took the locations in Southern Germany south of the Ruhr river plus the rights to those in the UK, Australia, and the US. Theo got Northern Germany and the rest of Europe.
In 1979, Theo purchased TJs as personal investment, not as an investment by Aldi Nord as that would have been in conflict with Karl's Aldi Sud's holding rights in the US.
Theo passed away in 2010, and Karl passed away in 2014 and their individual Aldi holdings are held in separate family foundations. Theo's individual holding rights in TJs are held in a separate family trust.
Thus, you have 3 separate and individual companies that have no common ownership, legal, financial, or operational relationships whatsoever.