Opinion is very split on this. I think when you dig down a bit it's actually quite a nuanced situation, further complicated because there are at least two separate issues.
In relation to this specific case, there is the issue of whether the ferry company either expressly or by implication assumed any responsibility for the welfare of animals on board.
Merlin's owner's interview with Beverley Cuddy says they assumed that they were paying for staff to check on dogs, not that they were ever told that this would happen. I don't think that is a fair assumption to have made, and I personally would not trust general ferry staff who may or may not have any animal experience to make sufficient checks or to be able to identify a dog in heat distress.
My understanding of dogs on these ferries is that they are "tolerated". As in you can bring them to leave in the car if you want completely at your risk. But the ferry company would really rather you didn't so levies a charge.
Against that backdrop I would not have left a pug in that car on that ferry. And I would not use that ferry at all. There are alternatives. Owners must then exercise their own judgement and decide whether to assume that risk. I don't agree with the premise of the petition that ferry companies "make" owners transport their dogs in unsuitable conditions because it is owner choice to use that service at all.
The second, further, issue is whether ferry companies should a) assume some responsibility for the welfare of animals on board and/or b) make provision for pets to travel safely and without suffering. My initial feeling is that the welfare of the dog should remain the responsibility of the owner throughout the journey but the ferry company should, if they are to allow pets to travel, make proper provision for safe and suffering free carriage of animals. If the petition centred on this rather than blaming the ferry company for forcing people to use unsafe means of transporting their dogs then I would support it. In fact Merlin's owner seems to be saying that she doesn't blame the ferry company either.
That's a bit of an essay, but without breaking it down like that then it's a very polarising debate.