I like dogs, but I have severe allergies. I don't find it easy sharing my space with guide dogs, as much as I respect and like them. Labradors are lovely, but they're not exactly short-haired and I can always smell them - but they are extremely well-behaved working dogs and trained to focus on their owner, which helps. There's no chance they would approach me, jump on me or lick me. Guide dogs are very uninterested in people who aren't their owner unless those people are doing something to endanger their owner.
Given the number of seeing dogs out there, it's rare I'm in an enclosed space with more than one guide dog, so usually I move away and I only have a mild reaction rather than a serious one. It does makes me ill, and there are consequences that last well after the dog and I part company, but it's a mild level of discomfort.
Sometimes when you have people with competing disabilities, you have to weigh up who is in greater need. It's safer for me to have mildly impaired breathing and days of itching afterwards than it is for someone who cannot see to have no help on a busy tube carriage where people are constantly getting on and off every few minutes and overcrowding there.
People randomly bringing their pets into the supermarket? I don't see how their need trumps mine. Service animals need to be specific breeds, trained up by people who know what they're doing, and used only where there is an absolute need. Severe allergies are actually considered a disability - something many people don't realise - because they have a profound impact on our lives.
There is a difference in how we react when one dog is nearby, compared to many dogs nearby, especially depending on the breed and how much they shed. A dog in an enclosed pram might not trigger a reaction in someone, but it might in another. You can't assume sticking a dog in a pram is OK.
If you are anxious about going into a supermarket, you have options other than bringing in a pet. There are mindfulness techniques, for example - coping techniques you can and should explore through counselling. You haven't explained why having a dog in a pram makes you more able to manage your disability, but as a dog in a pram is not going to be of any practical physical assistance to you, I can only assume it is for a mental impairment or neuro-difference, and whilst I'm not downplaying the importance of mental health, there are other solutions than dogs.
A blind person has much fewer options - what, you want them to rely on getting a human to accompany them each time? They're entitled to their independence. Guide dogs may cause some problems for some people, but they enhance the quality of life for visually impaired people so much and are in some ways, an extension of them.
Someone with severe allergies - what are they supposed to do if surrounded by a variety of animals? It's not as simple as taking drugs. You can't stave off an allergic reaction with a shed load of antihistamines, and frankly, why should anyone be forced to medicate due to someone else's behaviour? Having to put up with guide dogs is a reasonable compromise. Having to put up with other animals is less so.
The adjustment for a mental impairment or neuro-difference (a random animal) cannot trump the adjustment for a physical condition where the person can die (not being surrounded by animals in a supermarket). You have one animal - you are opening the floodgates for others.
I'm sorry, OP, but I think you are being massively unreasonable here.