Why are there so many canines in fine art?

yet "like a dog" is rarely flattering

That the dogs are just about always subordinates and subservient makes the relationship easier to work -and of course, dogs are happy to comply since it's part of their instinct to be subordinate in the presence of a dominant being (canine or person). Human to human is a little more difficult since outside of structured organization, we don't tend to play subordinate in most Western culture. You do see some cultures where men play an exceedingly dominant role (where women by multiple facets of tradition play the submissive role) and there is less man-woman strife, yet there is man-man strife since outside of structured organizations men don't automatically measure up and decide they are inferior or superior and accept and internalize that relationship (though some Eastern cultures do display deference at a superficial level).

Moms also tend to have a soft-spot for the momma's boys.