Xbox live party chat history

I struggle to understand why not, if Microsoft is storing this data then any data collected would fall under GDPR, submit an SAR for the data you require, if they refuse you could always go to relevant ombudsperson or seek legal advice on your information
rights. Alternatively you might be able to go to the ICO if the data falls under the The EU-US Privacy Shield scheme.

Unfortunately, you cannot access the chat logs in a party chat the only way someone would be able to access the chat logs if a person's account was reported for violating community guidelines and their account is up for being temporarily being banned.
But even then the person would still not have access to the chat logs.

Hello

Unfortunately, you cannot, remember that it is important that you do not lend your account, unless you have great confidence in that person, since it is your account and includes personal data.

unfortunately not unless it is something extremely serious like a crime or fraud you would have to contact the technical service by phone and they would tell you the steps you have to follow I suppose that the thing will not be so much a greeting

You may have been in game chat while you where in an Xbox Live Party. Check the features in your party settings to make sure you are in Xbox Live Party chat and not in game chat.