I have actually come to appreciate this approach, especially with Amazon, as I commonly use several accounts (employer, personal).
I regularly need to answer the question "What account am I signed in as" and if it is the wrong one, I need to switch without much hassle. After all, I usually only care who I'm logged in as if I'm going to make a purchase, and I don't want to accidentally have a personal charge come in on the company credit card.
For the business, losing a sale because a user abandoned the sign-out-of-wrong-account-sign-in-as-correct-account process is a real problem, and the easier and quicker that process is, the more sales they will have.
The approach described in this question is very effective for this use case, which isn't limited only to my scenario. Households with multiple users who share a computer can also benefit, for example.
A button that says "Logout" doesn't help, because if I'm logged in as the right user, I don't want to log out. Rather, in a short, human readable phrase, the system tells me who it thinks I am and gives me a means to quickly change identities.
Usable systems leaves little ambiguity in the user's mind about what the state of the system is. Who the system thinks I am is a pretty important thing, and this approach is one way of keeping the user informed while also giving a clear path to taking action if the system status is incorrect.