"Owe" and "own" sound similar but have nearly opposite meanings
Solution 1:
Based on the etymology of the Old English āgan that both words derive from, it seems that at that time it could mean either to own, or to cause someone else to own. Thus "I āgan a beer" - I own a beer, it belongs to me. "I āgan you a beer" - I will make you own a beer, I will make this beer belong to you. The words aren't all that different, then - "I owe you a beer" is basically similar to "I will make you own a beer".