Hello Jure! Thanks for your question and sorry for taking time with reply - I didn’t notice the question till now.
Name and Avatar should be set in Memberchat project settings. You need to have your Memberstack Custom Field ID setup for both Avatar and Name in Memberchat settings. For Avatar it should be an URL with image and name is just a string with name.
Also after you finish setup try to reset your API key for Memberstack in Memberchat settings. It will trigger update of all members’ data.
Hope it helps. Feel free to contact me in case you need more help.