This is where the concept of placing authentication factors in specific categories (what you know, what you have, what you are) becomes complicated. While these categories tend to be thought of as independent, they do overlap in certain areas.
Passwords are traditionally 'what you know' and I'd guess many people would always classify them that way even in this scenario where the user doesn't actually know them. Other people would say the treatment of the password in this situation has converted it from a 'what you know' to a 'what you have' factor. Some might even claim it fits into both categories simultaneously.
From the authenticating system's perspective it can't really tell whether you've memorized a password or not. It probably makes sense from a threat assessment perspective for the system owners to assume the authentication factor is being used in a traditional way, even if it might not be. But from the user's perspective they should probably assess the risk associated with how they actually manage their passwords.
All this to say that I don't think there is clear consensus on the answer to your question, but the majority would probably still classify the passwords as a 'what you know' factor. And I don't think most people would consider the use of a stored password as multi-factor authentication (MFA) without adding another factor besides the password.
UPDATE: With regards to the new concern you edited into your question, I don't believe that a complex password policy requirement necessitates that users will use a password manager. Most people do not use password managers (only 12% use them sometimes) despite any password complexity requirements on the Internet sites they frequent. There may be some correlation between users recording a password and extreme complexity/length requirements, but I'm not aware of a study showing this.
It looks like you're trying to make a case that relaxing a current password complexity policy will decrease the use of personal password storage. Your alternative appears to be an XKCD passphrase style system that primarily cares about meeting a minimum length of 16 characters. And while I agree that these passphrases can offer better security without unduly affecting usability, I'm not sure you'll make much of an impact on password storage. A study that compared use of passwords and XCKD style passphrases found that both sets of participants stored (e.g. wrote it down, had their browser save it, saved in password manager, etc) passwords and passphrases a pretty similar percent of the time. Both sets of users worry about forgetting their secrets, and take steps to prevent this. Even if a new policy allows passwords or passphrases that appear easier to remember than a previous policy it may not immediately change user behavior.
But back to your concern, while people may not agree on whether writing a password down changes its factor category, I haven't heard any serious discussion that this practice means a system relying on it as a knowledge factor loses MFA/2FA status when this happens. I believe most people would still consider it 2FA.
If you want to write up a justification on relaxing your password complexity requirements and advocating XKCD style passphrases I think there are better arguments for this change that don't count on eliminating password managers as a benefit.