Why do I have a picture on my contact list of a iPhone number I wrote down?
You can remove any contact photo from the contacts app. It’s going to be hard for us to guess your setup without sitting shoulder to shoulder and reviewing all the places you could have added an account. The likely culprit is the other person chose to share that image with you when asked by Apple permission to share their name and icon. They may not be aware this happens or mistakenly tapped the allow button while they were putting the phone in their pocket. It’s a one tap to share situation in some cases.