The name is Inbox App, not Inbox. Nothing special there.
I'm willing to bet that they'll have significant problems with the name if they try to call it just "Inbox". First, they didn't even get the inbox.com domain, theirs is inboxapp.com. Second, they don't seem to have a trademark on the product name as "Inbox" and I doubt that they'd get one. Third, their company name is Inbox App, Inc., so it doesn't seem like they did anything special in terms of buying rights or domains.
You are probably right about the company name, but inboxapp.com is still a pretty good domain name. Looking at the whois record for the domain[1] shows that it was registered in 2008. Moreover, archive.org[2] tells us that it was parked at GoDaddy for some time, before they purchased it in 2013. So, they probably invested a bit of money in a good domain name, yes.
Nice job on the research. You're right that they may have actually paid a premium for the domain. Though, I dislike domains with words like "app" in them as they tend to pigeonhole the product.
