I have installed my original version of Photoshop Elements 14 on a new computer. When I open the program on the new computer I receive the dreaded "The serial number is already in use by the maximum allowed computers. You need to deactivate another computer to use this product." I closed the program, and then logged into my old computer, opened PSE 14 and signed out via the help menu. Thinking this would do the trick, I then opened the program again on the new computer and was welcomed with the same message. I closed the program again, contacted Adobe through the help chat, and was instructed to completely uninstall PSE 14 from the old computer. I did this and opened the program again from the new computer and got the same serial number message. Please help, someone.
You did nothing wrong and you don't need to uninstall the program on the old computer.
Keeping it installed enables you to use your personal license by signing out of a computer to be able to sign in on another one by entering the license key.
If you enter the license key in your case and you still get the same message, that's not normal. That happens to users who have mentioned their key on a public forum for help, and the key has been pirated to be used on another computer.
Only Adobe Staff can solve the issue; insist that it's a licensing issue which needs to be taken care of by the proper Adobe Agents for licensing.