Can I delete the registry key under Software\Microsoft\Registration?

No, keep the parent key/folder please. But the parent Registration key is a shared Microsoft registry location and could be used by other software now or later.

Also, Software\Microsoft\Registration is not normally the primary location Windows uses to build the File Explorer “Open with” list. Those associations are commonly stored under locations such as:

HKEY_CURRENT_USER\Software\Microsoft\Windows\CurrentVersion\Explorer\FileExts<extension>\OpenWithList

and

HKEY_CURRENT_USER\Software\Microsoft\Windows\CurrentVersion\Explorer\FileExts<extension>\OpenWithProgids

In short - no - you should not delete the parent Registration folder. The Registration folder is a core Windows registry location. Operating system components and other active applications use it. Deleting the entire parent folder can break system features or cause errors. You might only delete specific subkeys that belong to the old app.

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hth

Marcin