Not entirely a fabrication, but not accurate, either.

As the other answers point out, he's saying “Daniel-san". “-San” is an honorific suffix, roughly equivalent to how we'd use “mister" in American English.

But this is a bit off for a number of reasons. First of all, like with “mister", the term is pretty much always used with the last name. So the more appropriate term would be “Larusso-san".

A more minor point is that “-san" is a relatively formal honorific, the type you'd use with relative strangers, or with elders. Given his closeness with Daniel, and the fact that Daniel was much younger than him, you might expect him to use the more informal “-kun" or “-chan" (though Daniel was old enough that “-san” is still reasonable, as a sign of respect.)

An interesting point is that, in traditional Japanese, the family name was used before the given name, so one could handwave that Mr. Miyagi thought that “Daniel" was his family name, and stuck with it even after he got to know him better. Problem with that is, he'd lived in the US for long enough that making that kind of mistake would be insultingly implausible.

And that's the larger issue about Miyagi's character. While he's generally treated respectfully, he's also excessively stereotyped as to what Americans think Japanese people are like. And the most notable example is his speech. Mr. Miyagi is established to have lived in the United States for over forty years, at this point, and his English vocabulary is perfect, yet he’s still given a deliberately choppy and stilted grammar. One can expect him to still have an Okinawan accent (though even that would likely fade over four decades), but the idea that his English would still be so broken after living most of his life in California is fairly insulting.

The point is, all of those Japanese flourishes in his speech are dropped in to give a foreign and exotic flair to the character, rather than out of careful accuracy. They wanted him to seem more Japanese to people who had only a rough idea of what Japanese people are really like.