Anybody can claim anything they want. In reality, should something go to court, it would quickly be revealed that Apple does not and cannot have a trademark on

Anybody can claim anything they want. In reality, should something go to court, it would quickly be revealed that Apple does not and cannot have a trademark on a word as generic as "pad". But what would you have for it? The court is probably not going to force Apple to put your app in their store. An antitrust class action might be plausible, but that's not what we're talking about when we talk about trademarks. It's misleading on Apple's part to call this a trademark issue.