You can assign F-keys to panels in Keyboard Shortcuts.
You can assign F-keys to actions in Actions.
But they know nothing about each other.

When you try assign a combination that is already taken by another command, Keyboard Shortcuts window will warn you and give ways to solve the conflict... but not if the F-key-combination is taken by an action — it knows nothing about actions.

Same with Actions panel — it knows nothing about Keyboard Shortcuts and never warns you, if you try to set, say, Shift+F8, which will disable native Transform panel call.

Please let these two work in the same database and know about each other and warn users.

Also, please vote for these related requests:

https://illustrator.uservoice.com/forums/333657-illustrator-desktop-feature-requests/suggestions/31039456-assign-shortcuts-to-custom-scripts-and-allow-actio

https://illustrator.uservoice.com/forums/333657-illustrator-desktop-feature-requests/suggestions/37343110-allow-custom-keyboard-shortcuts-for-custom-actions