So many sites that get a lot of traffic will have certain mechanisms to ensure the security of their users, as well as the website itself. A well known example used for similar purposes is the captcha. I'm pretty sure eBay has them. There was recently a bug fixed where it forced me to sign in again if I went from eBay dot com to the community dot ebay sub-domain. This is a rare example, and the first/only time I've ever encountered it. 99% of the time this is a feature, and a very important one.
When you are traversing a website and the site prompts you to re-enter your credentials, it is attempting to verify that you, the owner of the account, are trying to access the page and not someone (or something) else. The times where it is attempting to verify that you, the owner of the account, are the one attempting to access it despite already entering your login info is another important security feature referred to as Two-Factor Authentication and you should utilize this whenever offered.
These prompts to confirm user identity can be triggered in many ways. Many security implementations today utilize AI models that have been trained to discern user interaction from non-user automated input. This is just one example, as hopefully you might be a little more relieved. Logging in with two separate credentials might be inconvenient, but it is much less inconvenient than having someone drain your bank account.
Now about that first part....
I don't think anyone can give you a definitive answer to what you're describing in the first sentence. I will assume for the sake of it that this is occurring. First: this is highly unlikely to be a result of a "technical" issue on eBay's end. It just doesn't work like that. Either your credentials were leaked, or the issue is due to something that is local to your device or network. If you are seeing weird things like other peoples names within your own account go right to your purchase history. No matter the outcome you should immediately change the password on (at the very least) your eBay and email account. Different passwords. I recommend everything up to the router if you want to err on the side of caution. It is quite possible that two factor authentication you were complaining about is why you didn't just get locked out. Check all devices for software updates and after that shut them down and leave them unplugged from a power source for a few solid minutes. This will ensure that the cache in your RAM is cleared and wont boot up with any data stored there from the device's last session.
If you aren't necessarily sure what happened, doing the above just to be safe is never a bad thing as long you don't forget the logins, and store them securely. You are likely fine as long as you change your password. If you do see anything suspicious on your account you should call or report it to eBay.
Hope this helps.
