Ex-chicken farmer here... 4-H, APA, all that rot.
This sort of thing is easily possible and, in fact, not too uncommon. The egg's development is in stages, starting out with a yolk that leaves the ovaries and passes through the oviduct. It then gets "plated" with the egg white in the infundibulum, and then the shell is attached later, starting in the magnum.
Now, once in a while -- maybe 1 in 1000 -- the egg can accidentally reverse its flow. If it passes back into the infundibulum after getting a shell, the complete egg will acquire a second layer of white, and then a second shell.
Another not-too-uncommon occurrence is a double-yolk egg. In this case, the second yolk catches up with the first, and both get a common white and a common shell. This is more typical. A high-producing fowl can crank out nearly 300 eggs in a year, and some average more than 1 per day! Lots of eggs on the assembly line at any given time.
The egg in this video is unusual in that it is
both a double-shell
and a double-yolk. I never saw one like this, but the mechanics are the same. I find it to be totally credible.
The other oddity one sometimes encounters is a "shell-less" egg. Sometimes the egg doesn't get a proper coating of shell, particularly if the chicken is calcium-deficient, but the shell membrane is enough to hold things together. This results in an egg that looks and acts like soft rubber.
ETA: Something for the lady chickens...
