In the original Muppet Movie (lyrics, video), Jim Henson asserted there is not, but that was 1979.
Now it's a few decades later, and the English language has evolved, including a lot around changing speed and depth of new relationships. It seems possible that there might be a word that describes a first meeting of two people that is accompanied by a resonating familiarity, déjà vu, and sense of overlap in perspectives and values, leading to a feeling of a strong relationship despite the absence of actual length.
Also in the past few decades, we've gained the EL&U Stack Exchange, where such a question can be asked. So, give it your best shot!
The target word could be a noun describing a person (with grammatical usage similar to other relational terms like friend, enemy, acquaintance, companion, etc.) or the feeling (with usage similar to déjà vu but focused specifically on a person and indicating a feeling of strong connection but not necessarily a confusion over having actually met the person or interacted extensively with them before). Upvotes would also be given for words that describe the experience/encounter/scene (with grammatical usage similar to meet cute).
Kaukokaipuu, "a feeling of homesickness for a place you've never visited," is something that may be reduced/resolved by the event of meeting this person, although with kaukokaipuu it seems like you know in advance what it is that you're missing. The book with that word may also have an answer to this question.
Best Answer
In a search, the closest I could find is "Instant Rapport," presumably described in a book of that title by Michael Brooks published a decade after the song quote adapted for this question's title.