What is AEServer and why is it requesting macOS Accessibility permissions?

What is AEServer in the macOS Accessibility permissions panel?

System Preferences > Security & Privacy > Privacy > Accessibility > AEServer

Why is AEServer requesting to control my computer via accessibility APIs?

Should Accessibility permission be granted for AEServer?

As benwiggy already said, AEServer is a background process that processes incoming AppleEvents from remote Macs.

AppleEvents are the underlying technology used by the Apple Event Manager, which is in turn used by e.g. AppleScript to communicate with applications and processes.

Starting with macOS 10.15 Catalina, remote AppleEvents have additional security requirements: The user account sending the remote events has to be the same as the one receiving them. More details in the release notes.

To turn off this restriction and revert to pre-10.15 behavior, you can use the following Terminal command:

defaults write /Library/Preferences/com.apple.AEServer RestrictAccessToUserSession -bool false

AEServer is the Apple Events Server. It is used to control Apple Events that come in from other Macs.

Check System Preferences > Sharing to see whether Remote Apple Events is turned on.

As to whether it should be running and given permission: that depends on what you do with your Mac.

Do you control this Mac from other Macs?

If you turn off Remote Apple Events and don't grant AEServer accessibility permissions, then I dare say something will complain or not work if it needs it.