We all know that the 1972 Dolphins didn't lose a game the entire season and are recognized as the only team to do so (not counting New England since they lost their last game, I'm aware they didn't lose a regular season game). But what about the 1929 Packers, they went 12-0-1 winning the NFL title. Could they be considered as the first team to accomplish this or did they not accomplish it because of the one tie.
I think they are talking about unbeaten and untied teams.
Technically the 1934 Bears were unbeaten and untied in the regular season with a 13-0 record but lost the NFL championship game to the NY Giants 30 to 13.
Oddly enough the 2007 Patriots as you mentioned were unbeaten with and untied a 16-0 record, won two playoff games to get to the Super Bowl and also lost to the NY Giants.
The 1948 Cleveland Browns were unbeaten at 14-0 in the old All America Football Conference and won the championship over the Buffalo Bills 49 to 7 for the first Pro Team to go unbeaten and untied for the regular season and playoffs. The Browns were a legit great team but it's debatable whether they could have gone unbeaten in the NFL.
A few teams have come close. The 1984 49ers for example were 15-1 in the regular season and lost a close game to the Steelers 20 to 17. The 49ers won three games in the postseason to finish the season 18-1.