It is possible to have one account signed in for general use of a device and a different one for app and media purchases. It appears that is what you have done.
If you don't know the password for an account and you you cannot meet any of the alternative identify confirmations being offered by Apple you will have to apply for account recovery. You can request account recovery as in this article: How to use account recovery when you can’t reset your Apple Account password - Apple Support Account recovery might take a few days or longer, depending on the specific account information you can provide to verify your identity. One tip to using account recovery it to look for options in small blue letters beneath the normal log-in items if you cannot complete the requested information.
Here are links to posts by Community contributors which provide additional detail about the account recovery process and what to do if you do not receive a text or call: Activation Lockout Issues - Apple Community and How can I reset my Apple Account password? - Apple Community and How to delete Apple ID when you can't log… - Apple Community This thread outlines the multi-step process and final recovery —> My Apple ID is locked, how do I fix it? - Apple Community , and this one about things you should and should not do while waiting —> How can I recover my Apple ID after multi… - Apple Community
If you cannot recover the account, you may at least be able to regain use of the hardware again for use with a new account. Refer to How to remove Activation Lock - Apple Support for the various ways to remove an Activation Lock. If you have the necessary information you can do this yourself, otherwise Apple can completely reset a device if a person can provide sufficient evidence of valid ownership. Check the requirements and steps at: Activation Lock support request - https://al-support.apple.com/#/additional-support Apple says, "Proof of ownership documentation is required," but does not detail what currently constitutes proof other than it “must include the product serial number, IMEI or MEID.” Apple may typically require the original receipt* issued by Apple or an Apple Authorized Reseller. If Apple resets the device this will give you access to the device but won't restore access to the account used to lock it. There is nothing that can be done to expedite an Activation Lock request.
*Apple recommends you "Keep your receipt after you buy an Apple device" --> Keep your receipt after you buy an Apple device - Apple Support
