Is it a statement of her ownership, and she wants people to know that he is owned ?

Uh so they dont run away

We have ordinances and laws here that a pet must be kept under control when outside and not fenced in. It is in place for public peace and safety as well as the safety of the animal. Most places require that they have their proper shots (like rabies) and are licensed...a collar makes it easier to put a leash/harness on them as well as keep their tags with them (as well as identification should they become lost or stolen).

So they can be easily restrained when needed and most towns require it.

For their own safety...
And others....

Mine is an indoor dog so he doesn't wear one.

Not all pets are obedient enough to follow all your commands.

Because of the law... same as chipping pets...

Because she likes it.

We collar our pets so that other people know they are owned and don't simply turn them over to the dog catcher for euthanasia.

Next question.

The same reason we collar our sexual partners..

Because, strictly speaking, we are responsible for them. If they catch a scent and are tempted to investigate it further, having them on collar and lead prevents them from dashing off into traffic or other danger. We can’t exactly reason with them as to why they shouldn’t go.

So they can be tethered when it's safer for them or others and so they can have identifying tags in case they get lost.