"I don't think that buy.com was well established enough for Rakuten to stay with it. Was there a lot of customer loyalty to that name?"
36th in the top 500 -
http://www.internetretailer.com/top500/list/
"They buy chains, such as Asda in Europe, or Seiyu in Japan, and don't immediately switch over to Walmart."
Yep
http://www.ehow.com/info_8425002_walmarts-name-different-other-countries.html
"Walmart's first international store was a Sam's Club in Mexico. Today, other Walmart store names in Mexico include Bodega Aurerra, Bodega Aurerra Express, and Superama. The company's other international brand names include BestPrice Modern Wholesale in India, Asda Supercentre in Great Britain, Seiyu in Japan, and TrustMart in China."
"In most cases, Walmart decides to keep the name of the local retailer so shoppers will be familiar with the brand."
That's my point. People here are already familiar with the name.
"To date, though, I can't think of a single giant company that used a generic to get there. I know that's sacrilege to some domainers, but I think there are limits to how far a generic domain can take you."
That would be more for pure online type merchants, such as Buy.com. Most big merchants existed before the internet, so they use their established brand. Then you have some that came about with the internet - Buy.com, Shoes.com, Hotels.com, Art.com, Cooking.com etc., all on that top merchant list. While they have a generic keyword, they've built it up as a brand.
Who knows, looks like they've done it successfully in Germany, rebranding Tradoria and Brazil rebranding Ikeda. So you can find examples of both strategies working.
But when you read comments about this, it's typically what you read here -
https://www.appeagle.com/Buy.com-changes-name-to-Rakuten-shopping
Pretty negative to the name change. You also have to consider some Americans are not exactly fans of Asian countries buying up American companies, sometimes there is a little backlash. Might just go to Amazon.