For starters, it's not practical to leave animals at home if you're (a) moving, (b) giving the animal to a new owner, © taking the animal to a veterinary appointment,...
I suppose some here would like everyone who needs to do any of this to rent a car or fly. I prefer to promote train travel, myself.
Half the time the pet is the unhappiest one on the trip. Confined, unfamiliar sounds and smells, etc. Just so the owner can have that "unconditional love" dragged around. Get a tomagotchi.
Working dogs are completely different from pets.
I would never take my cat on a train ride, it would be sheer misery. Cats are territorial animals ... much happier at home than traveling like that. Like many pet owners I trade pet sitting/check-in services with other pet owners, we trade keys so our pets are safe and happy when we travel. You can also pay people to do this, not super expensive, actually.