So, this person is very open, honest and brave to display his vulnerabilities and feelings (because that's what feels natural for him, and as a good thing, to do) whether he's out in public, with strangers, or not. He's being particularly open about his feelings and vulnerabilities, even when it may not be asked for or necessary to do so. There is nothing fake or deceiving about this person. You're always directly connected to whom this person really is.
A transparent person may be a possible definition.
easily understood; manifest; obvious.
candid; frank; open.
Like a looking glass into our soul. Often it is something that isn’t done. It is a way to keep our true self from being seen by others. Being transparent and showing your inner light to others is not something everyone does.
A genuine person fits also.
free from pretense, affectation, or hypocrisy; sincere: a genuine person.
It can also be applied to personal attributes like feelings, sympathy etc.
Someone who does not hide their feelings is said to "Wear their heart on their sleeve", or for a single-word description, the person can be said to be "guileless", meaning there is no insincerity or pretense to them.
As defined in the Google Dictionary:
Guileless: "devoid of guile; innocent and without deception."
While its antonym, disingenuous is much more commonly heard, you might consider ingenuous
candid; frank; straightforward