That is just a bit of bad reportage.
McConaughey customized a trailer with Timeless Travel Trailers, which was based on an Airstream shell but so highly customized that it is legally registered as a "Timeless Custom," not an Airstream.
He also separately customized a 1953
Airstream Flying Cloud with GSM Vehicles. This was partially documented in Airstream Life, as "Vintage Lightning," but I hear Matthew calls it "the canoe." (Incidentally, GSM Vehicles has changed ownership since that job was done, and Colin Hyde, who envisioned the project, now runs his own restoration shop.)
The reporter seems to have read a few inaccurate reports on the Internet and mashed them up to create one mythical trailer.