Repeated sign-in requests are often the results of a browser's inability to update cookies.
Check your eBay account site preferences to see if the "keep me signed in" box is checked. Sometimes unchecking, applying, and then rechecking the box and applying again can help.
Disabling ad blockers, pop-up blockers, script blockers or other privacy tools on the sign-in or verification page can help; many of those have settings that can interfere with verification. You may be able to turn them back on once you have signed in and updated your eBay cookies.
If you are using Firefox, try disabling Enhanced Tracking Protection on eBay using the shield icon to the left of the page address.
If you are able to sign in using a private or incognito browser window, that might indicate a problem with a corrupt or blocked cookie. Try erasing just your eBay cookies manually in that case.
If you are routinely clearing all your cookies, you will likely have to go through verification again each time you sign in.
Or try a different browser -- preferably the latest version of a popular browser that you do not normally use and one that has no add-ons or extensions installed.
