This is expected behavior for third-party browsers. When you use the 'iCloud Passwords' browser extension, you need to enter a six-digit verification code to unlock the extension (it's not a 2FA code). When you quit Chrome, the extension is locked and you'll need to unlock it with another verification code the next time you use it. The only workaround is to just leave Chrome running so that it remains unlocked (or switch to Safari).
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