Safari keeps asking permission to access the keychain

Yes, if you press Always Allow would put it back to it's default behaviour otherwise it will ask you to access the Keychain each time. By pressing Allow you tell the security system to allow Safari to access it once and only once, if you say Always Allow you say it can access it from now on until it is updated again. (Developers have an option to avoid this issue by signing their applications)

"Though given your description I doubt it's the case, but maybe the digital signature of Safari itself is broken? You can test it using:

codesign --verify --verbose /Applications/Safari.app

...which should give you:

/Applications/Safari.app: valid on disk
/Applications/Safari.app: satisfies its Designated Requirement

(And if indeed it's broken, then see If Mac code signing is tampered with, what might fail?)"

"It may have also become out sync with your login account. By default regular keychain first aid won't fix it. To make it check and fix it, open Keychain Access.app, click the Keychain Access menu and open preferences. Hit the 'First Aid' tab and tick the 'Synchronise login keychain password with account' box.

Then run keychain first aid again."