I am a caregiver to someone who has a service dog, with

So you are not the first one to have this issue, but non-professionally trained service dogs are not considered to be "service dogs". They are therapy dogs or companion dogs, which may or may not be protected in the human rights code.

Oh, I am sure they are well trained, they are just not covered by law if there is no certification.

To start off, service animals are governed by various pieces of legislation

1) Blind persons rights act

2) AODA (if you are in Ontario, if not then your province will have a similar piece of legislation)

3) Human rights code

Each act defines the service animal differently, the widest one is the human rights code which states that a service animal is one that is "physically relied upon".

You are welcome to test this bring bringing an action against Air Canada in the human rights commission to make a determination as to whether or not their denial of your animal is discrimination and failure to accomodate. In all cases of accommodation, the request could be necessary for the individual but the accommodation could amount to an undue hardship for the organization.