Doug, quite simply, lives for the job, not so much for Frank. He knows what he’s good at, and doesn’t step outside of it. He’s loyal because loyalty is a prerequisite for anyone doing the kind of work he does. Not being capable of loyalty would mean nobody would trust him with their deepest, darkest secrets, and without trust, any would-be chief of staff is done.
So now we know why Doug is loyal at all, but why Frank? Frank is the perfect vehicle for Doug. He’s the 90s Chicago Bulls, playing for any other team would be a huge step down for Doug. Politicians are in it to win it, and Frank’s a born winner. Also, once you’ve worked with someone for so long, starting all over with someone else is like throwing away a 20 year marriage. Sure you can do it, but it’s bound to be painful and you don’t want to pull that trigger unless you absolutely have to. Dunbar was a complete fool to have trusted Doug as little as she did.