The obvious thing is that all these teams were crowned champions in that year. The link is related to that.
They all won the league two seasons after being promoted. Bill Shanklys Liverpool finished 8th in their first season back in the first divison in 1962/1963.
Its nothing to do with the communinty shield as Liverpool won it outright in 1964 but that would have been the start of the 64/65 season. Liverpool were in it again in the 65/66 season where they shared it with Manchester United.
Liverpool were crowned champions in 1964 but the charity shield is played at the start of the following season. Manchester United were champions in 1962/1963. They won the community shield in the 63/64 season. Liverpool champions in 1964 won the charity shield in the 1964/65 season and shared it as FA Cup winners against league champions Manchester United at the start of the 1965/66 season. So in a nutshell, its can't have anything to do with the charity shield.
I'll put you out of your collective misery. They all share the dubious honour of amassing the most number of losses for a top flight championship - 11 defeats
Liverpool hold two other dubious records. They hold the record for the highest number of points for a team that finished runners up(86). Thats 7 more points than what they won the league with in 1990 (79) They are also the only side to have only lost twice and not win the league. Both 2009.