To check your other eBay accounts, use a new browser, or a private or incognito browser window. You can stay signed in to multiple account if you use one browser for each account. Go to eBay's sign in page and enter the username and password of the account you want to use in that browser.
If you have the eBay app on your phone, it may be set to open eBay links in the eBay app automatically. This can be a problem if your app is signed in to a different account than the one you are trying to use. You may have to explicitly sign out of your eBay account in the eBay app if you want to use the app.
Or, to prevent eBay browser links from opening in the app automatically on an iOS device, tap and hold an eBay link in a browser and you should see options, one of which should be "do not use the eBay app to open this link" or something similar to that. Your choice should be remembered the next time you use an eBay link in that browser, and that should allow you to use eBay in a browser without automatically opening the eBay app.
A factory reset should erase all your personal data on the device. But if you have your Apple account backed up, restoring from backup may simply restore all the data you just erased, including cookies. You may have to turn off your backup/restore program to prevent that from happening.
One further possibility if you want to sign out of eBay on every device. Try this:
Go to your eBay account settings using a desktop browser (or a mobile browser in desktop mode). Under sign-in preferences here:
https://accounts.ebay.com/acctsec/security-center/signin-preferences
Unlink any social media accounts: Google, Facebook or Apple from your eBay account.
Uncheck the "Keep me signed in" box at the bottom and apply changes.
Finally, change your eBay password.
That should force you to sign in again in every device to use that eBay account.
