There are limits to the number of times you can sign in and out per day using the eBay app. It sounds like you may be running into those limits.
Try using a browser to go to ebay.com rather than using the eBay app if you are running into this error.
You might be able to avoid signing in and out so much by using the app for one eBay account, and using a different browser for each other account. Or a different private or incognito browser window for each account. That way you can stay signed in to multiple accounts at the same time and quickly check messages or sales without needing to juggle sign-ins all the time.
