The line dividing Germany into Aldi Nord and Aldi Süd territory — https://upload.wikimedia.org/wikipedia/commons/f/fc/Aldi_equ... — is called the "Aldi equator". We sometimes jokingly call it Germany's most important dividing line post-reunification :)
tl;dr Aldi and Trader Joe's are (theoretically) in competition, and neither owns the other. The other Aldi owns Trader Joe's.
The two Aldi locations I drive past are both located less than a mile from the nearest Walmart locations (I expect they do a fair amount of piggybacking).
The purchase doesn't make a huge amount of sense to me. I guess the most sensible thing is that they see the stores being located by the best populations to start out doing food deliveries for (affluent, so they can charge higher prices while they figure out how to do it).
Walmart has a pickup service. They keep experimenting with rapid delivery but people don't really want to pay much for it.
1. http://www.chainstoreage.com/article/different-kind-drone-ro...
If this works out for Amazon, I wouldn't be surprised to see Walmart try and buy Trader Joe's. They own Aldi, which has made a big push to compete with Walmart's reach outside cities. The nearest Aldi is closer than Walmart, and that's already not far away.