What do you mean by, "important"? That it's impolite to not refer to someone using their preferred title, or that people tend to expect them?

Depends on the person, some people do get insulted if you don't use their title in a formal setting. As said, it's rather silly IMHO.

Never had a Herr Doktor for a supervisor huh?

A doorbell I can understand;[1] it's your own home. But the requirements might be lower if you are signing up for some Lidl customer benefit club.

[1] Well, personally I think that's silly too. But it's a question of degree.

Can confirm. In Germany, recipients of a US visa often complain that their title isn't used on the visa, only their name. Especially PhD visa applicants.

I think that's just out of the necessity of not having to explain who exactly "her Dr Rer Nat" is.

My mom listed our family as my dad's name with "Dr." in front, I think in the phone book. My dad was never happy about that. But we're not German.

not to mention that often the title is Prof. Dr.

We had a go-live delayed because the sponsoring executive needed to demo the app to a board member, and would not do it until the UI showed the full greeting of “Prof. Dr. xxxx”. We had to shoehorn that into the data model at the last minute.

Yeah, professor is an actual title here, you can be a professor without being tenured.