It’s a popular question, and the answer is no. They do have a shared history, though.
Aldi opened its first store in Germany in the early 1900s. After World War II, the owner’s sons took over and expanded the business into one of the largest grocery chains in Europe, with hundreds of locations. In 1960 the brothers had a falling out and split the company into two separate entities, Aldi Nord and Aldi Sud.
The two companies still exist today, operating stores in their respective regions, both under the Aldi banner. In the interest of consistency, they cooperate where they have to.
Aldi Sud founded the first American Aldi store in 1976 (in Iowa), and currently owns all U.S. locations. Completely unrelated is the fact that in 1979 Aldi Nord purchased the U.S. operations of Trader Joe’s, which was founded in California in 1958.
So, there is no relationship between Aldi and Trader Joe’s.