"The thing is, is that..."

In the linguistics literature, they're referred to as either "double is" constructions or "thing is constructions". Various papers have been published on the subject which you should find if you Google these terms.

In one possible analysis, what is happening is that there is effectively an element such as "what" which is implied:

"(What) the thing is, is that I'm going to be late"

"(What) my feeling was, is/was that..."

Note that although they're referred to as "double is" or "thing is" constructions, the construction doesn't necessarily involve either a double "is" (notice "...was, is..." is possible) or "thing" (notice the example with "My feeling").

I'm not aware of a claim that the phenomenon is restricted to a particular dialect.

This is incorrect grammar. However, it looks like something which was spoken out loud: people can not delete or edit their previous utterances, so people often construct sentences verbally which are not strictly grammatical.

It not only sounds wrong, it is wrong. And the introductory words are not a phrase, but a clause that already contains both subject and verb and so does not require a secondary, redundant verb.

"Double is" is not a sufficient name, since the usage may not include any form of the verb "is", such as "The problem being, is that ..."

It is correctly called a "double copula" (a copula being the linguistic term for a word used to link the subject of a sentence to a predicate).

According to the third edition of Fowler's Modern English Usage (as revised by Robert Burchfield), the double copula originated around 1971 in the United States and had spread to the United Kingdom by 1987.