We think of dogs as faithful companions; hence the use of the word 'Fido' as a generic term for domesticated dogs. Google Answers result for this name says that Fido was Abraham Lincoln's dog and because of Lincoln's celebrity status many people gave the same name to their dogs.
It certainly seems that Lincoln valued Fido, even though he couldn't take the dog with him to Washington.
I wondered why this name ended with the letter 'o' and then I recalled from my long-ago, short stint of learning Latin that this is a first-person verb ending. A site listing male names says it means 'I trust'. Hmm... who trusted whom? Did Lincoln name the dog because he trusted it, or did he believe his dog trusted him?
The University of Notre Dame Latin dictionary defines 'fido' as 'to trust, believe, confide in'. Okay, that's good - next time I find myself confiding my problems to my dog I won't feel embarrassed.
Etymology of First Names defines it as 'I am faithful'. Seems to me that's putting a slightly different emphasis on the relationship.
Eric Partridge, in Origins, A Short Etymological Dictionary of Modern English, defines this name on page 197 as having probably come into English through the Italian word fido, meaning trusty.
So: we trust dogs
or we expect them to trust us.
Either way it's an important relationship.