Domestic tags are generally for docile, outgoing, or friendly personalities and low-danger species, and indicate a Companion that is unlikely to hurt you. For example, Esmond may look rather odd, but his Domestic tag lets you know he's really quite friendly!
Stray tags are typically for more introverted, withdrawn, or subdued Companions. They don't lash out often, but may be more difficult to raise than a happy Domestic. Strays aren't usually outright dangerous unless pushed (just like anyone else). Alexi is a good example of a Stray Companion - a Sea Slug with a bit of an attitude.
Wilds are for Companions that are either physically more capable of doing someone harm (poisonous, long claws, sharp teeth, big scary animal species, uniquely dangerous in some manner), have more violent/angry personalities or tendencies, or some combination of these traits. Wilds are generally considered harder or more dangerous to raise than Domestics or Strays. A good example of a Wild is Ashanti!
At adulthood Companions are considered genetically stable, and their collars may be removed.
If you would like collar art for a Companion character made post-move, just ask!