A little while ago, I wrote an article on LinkedIn regarding my own mental health issues. I thought it was important to be part of the movement to make these conversations normalized.
Normalized or not, this is a difficult topic for managers when it comes to the team.
First, managers are not - nor are they expected to be - therapists. Do not try to solve or address mental health problems. Your role is simply to do what you can to support them. Give them time to address the issue with a professional, mention any resources (like EAPs) that can help, and ask what you can do to help.
Second, realize it can still be a stigma for some people. Respect the wishes of the person if they want you to keep it private. It is simply a case where if anyone asks, you say you are respecting the wishes of the person as you would with anyone and will not talk about it.
FYI - notice "will not" not "cannot". You are making the active choice. Be assertive and own the choice.
Third, be honest with the individual. If they need accommodations and you can reasonably provide them, do so. You are required by law to do it, but that isn't the reason. The reason is that you want to care. If the accommodations aren't reasonable, then be honest and try to find a way to work around it. Obviously, work with your HR team to deal with the issue appropriately. Be cautious that some HR teams are way too strident about things and are so afraid of getting sued that they actually make the relationship between you and the employee worse.
Fourth, and most important, be human. This is a person struggling with a difficult situation. They can get out of it with the right help, but simply let them know that you care about them as a person and the work will figure itself out. You can't make any guarantees other than you will always be honest, fair, and helpful.