Lambeau Field may have a sellout streak dating back to its construction, but the Packers as a team don’t, which is why we Packer fans have to accept the fact that the Redskins are the sellout streak kings in the NFL.
For college football, Nebraska has a current streak of 303 sellouts. Notre Dame is at 212 - and if it wasn’t for ABC moving a game to Thanksgiving in 1973 when the students weren’t in session, it would be 251.
Ding! The Lambeau Field consecutive sell-out streak goes back to 1960, which would pre-date the Redskins’ streak…but we have some of those pesky non-sellout Milwaukee games in there. 
And, getting back to the question of, “is it the same people at every game”…sort of.
We don’t use our season tickets for every game; we sell a lot of them. (And, if you have Packer season tickets, you either have the “Green Package” which is largely the original Lambeau season ticket holders, and you get 6 regular-season games a year, or the “Gold Package”, which is largely the old Milwaukee season ticket holders, and you get 2 regular-season games a year.)
I usually go to one game a year, and my father goes to one game a year. We sell the rest – particularly the Bears and Vikings games, as those sell for the most, and pretty much pay for the rest of the tickets. We sell them through a service which works directly with the Packers, and indemnifies us (so, if the yobbo who buys our tickets gets drunk and assaults someone, they won’t strip our season ticket rights).
Old joke : little old lady is by herself at a big sold out FB game, the seat next to her is empty. Someone asks why the seat is empty. She says “my husband used that seat and he died” then they ask why did she not bring a friend or relative ? She replies “They are all at his funeral”
Nitpick: It was the Miami Dolphins.
The Toronto Maple Leafs sold out every game from 1945 to 1999, so this list cannot be correct. They also currently have just as long a sellout streak as the Wild do.