Source: I have a space in my name and some of my different identity documents have the name as "<A> <B C>" or "<A B> C", which causes all sorts of administrative problems.
Nowadays the meanings of our last names have largely disappeared, so you have countless Coopers who have never touched a barrel in their lives, whose children will be called Cooper also, despite that. I think it's a little sad that so much of what people call us is semantically equivalent to a random UUID with tons of namespace collision. With that in mind, I'd say "da Vinci" is more a last name than most of us have.
I can understand that in some cultures this might seem weird or antiquated but here in Germany these names are reality. Sometimes people with such names are descendents of royalty and sometimes someones last name "from family-name" is thier last name and happens to historically correspond to one of germany's state names or city names or just a little town.
One a side note: In Germany in 1919-1920 royalty was no longer a legal aspect that changed how laws applied to you[1]. When that happened titles that were reserved for ruling functions (king, grand duke) were removed and all other titles were moved to be part of the persons name (such as prince etc.) and could not be decreed on anyone new. These titles still exist in Germany but are simply naming "conventions" in a formerly royal family.
[1] https://de.wikipedia.org/wiki/Adelsrecht
[edit] In Leonardo's case perhaps not but still i wish to elaborate a litte on the situation here.
Seriously, though, it just feels apropos.
A UK passport has a "surname" field and a "given names" field. But a recent UK birth certificate has a single name field in which the "surname" part is distinguished by being written in capitals, so "Peter James ADAM SMITH" would have a surname consisting of two words. But what if a word consists of a single letter? For example, some Irish surnames look like "O Briain", and I think there is a Vietnamese name that consists of a single vowel, so presumably you can't always tell from the birth certificate which part is the surname.
The legal name however consists of First and Family names only, without the other names. Therefore many people have two names in the First Name field, usually separated by a space. The disadvantage is that in all official forms they have to spell out all the first name(s), no omissions or abbreviations are generally admitted.
Typical Portuguese names have around 5 names, two first names and three surnames from both parents.
In fact your description fits quite well how they are used in Portugal, plus a few other nuances.
So, Heinrich August Schmidt has a last name / family name (Schmidt) and two first names / given names.
*"you are" meaning you'd be a German citizen with a German passport.
When you marry, you get to define what is going to be the family name and then children born from that marriage get to be registered with that family name.
In the case of foreign-born people, they can keep the naming rules from their original country. In the case of the marriage between foreigners from two different countries, you have to choose which rules are you going to follow, but the family name stays fixed.
To us (Brazilian marrying a Greek) it was a very interesting process. I have two last family names, and Greek names are gender-conjugated (i.e, the last name changes whether you are a boy or a girl). It the end the simplest thing to do was to just keep only one my last family names.
This is possible because you can make a “name declaration” where you choose to apply the naming law of another EU country.
I have two middle names; most financial service providers decline to recognise my second middle-name. Same goes for the taxman and my pension provider. It seems that a "full name" is no longer a canonical identifier for a person, just really a kind of nickname; the canonical identifier is now an account-number, employee-ID or whatever.
It sounds as though they allowed the middle name to contain a space, which doesn't match my mental model but perhaps that's how they do it in California. It invites the question: do they allow a first name to contain a space, so that ("X", "AE A-XII", "Last Name") and ("X AE", "A-XII", "Last Name") are different names?
EDIT: The funny thing isn't allowing spaces in a middle name; it's have a separate field for middle name(s) at all. Modern passports have just two name fields: "surname" and "given names". Both may contain spaces. But according to images on the web, Californian birth certificates really do have three name fields.