I have been actively involved with Airstreams for over 15 years and have never seen a chair like this, so I am highly doubtful of the claim.
I looked up the name of the company on the tag that can be seen in one of the photos (ICU USA Corp.), and that company was dissolved in 1984 after 35 years in business, so we can thus put that age of the chairs before 1984:
Since the ICU USA address on the tag lists a zip code, we can put the earliest possible date as after July 1, 1963
I agree that from the look and materials, the chair feels more like 1970s or early 1980s than 1960s as claimed in the link you provided. I found several other websites that also put the age as from the 1970s.
My gut feel is that this is just someone trying to use the name "Airstream" to try to increase the desirability of the item, but there is a small chance I am wrong. I can be pretty sure these were never provided as original equipment with new Airstreams. What I cannot rule out is whether they may have been an accessory available thru the Wally Byam Store catalog that was in place from the mid 1960s into (I think) the mid 1980s.