I wouldn't be too sure. There is, unfortunately, a lot of room for interpretation with this law. Rather cautious adviser urge you to have one as long as your website isn't private in the sense that can't be accessed by the public (i.e. a website for family members that can only be accessed with a password).
This law (IANAL) is just horrible.
The law just requires a way to contact a person within of a few days, and a way to be able to get the person behind a site to be able to sue them.
The alternative is sites that you can't do anything against if they act illegally against you, e.g. through libel and slander.
If I want to send an Abmahnung to a website owner, because they publicly defame me — as has happened before — then I need something stronger than an email address. Or I need to invoke the right to be forgotten, but I don't want to do so, I prefer having a way to eliminate the source.
I was always wondering if they overturn this at some point. I don't know of any other country requiring it and I doubt that it helps taking down illicit content. If anything, it helps consultants for offshore companies.
I'd be fine with depositing my personal date somewhere so that people with legitimate concerns can contact me but it should require some effort so that the function can't be abused.
And as a reminder, the German rule only applies to commercial sites — a site with no ads and no way to pay such as quasseldroid.info does not need an Imprint.