Brighton are the only team that can affect my mood
I'm pretty confident that most of those, who consider themselves to have a 'second team', are refering to either a foreign side, or a non-league local team, rather than to Liverpool or Man United.
Or Brighton.
I have a soft spot for some other teams, but on a scale of about 1% of what I feel for the Albion.
Soft spot for Ronda CF. My family's Spanish "local" side, saw one of the most entertaining and bizarre games of football ever there a couple of years back.