There are several ways to personalise live chat starting from using your own name and picture when chatting. Another way to bring a personal touch to live chat is to try to express emotions, like empathy when chatting. Making the customer feel valued by using some simple key phrases like: “I appreciate your patience” or putting yourself in their shoes by saying “You are absolutely right. That would frustrate me, too”, will help you go a long way.
Although it is good to keep chat messages short and simple yet polite, communication style in live chat can still be quite casual. So getting personally invested and using your own tone of voice is also more than ok.