I was thinking about this the other day. If your bird is well behaved then I think it's a good idea, especially when going on vacation so you don't have to board or find a pet sitter. Same thing for dogs or any pet, who is to say they're not a therapy animal and if they cause no problems what's the harm?
If a medical professional cannot verify a legitimate service the animal provides, it shouldn't get the same rights as a real service or therapy animal IMO. If people continue abusing that system, it is going to eventually have an impact on people who actually NEED their service/therapy animals. We used to have a neighbor who lied about her dog being a "therapy animal" to rent in a no dogs/cats complex. Management couldn't question it or ask for proof either, since that's against the law. That dog was no therapy animal, it was a complete nuisance to everyone and took a year before management could get her evicted. It has made me question the legitimacy of service animals ever since, which no one should feel they need to do. You want to trust if someone is saying their animal is a service or therapy animal that it actually is.
People like that woman (the old neighbor) are making it very hard for people with legitimate service/therapy animals who do have special training OR like the OP has done, had a healthcare professional who can verify the benefit the animal provides the person, to keep those special rules surrounding their animals being allowed everywhere with them. Lying to sneak a pet in somewhere, even a well behaved one who isn't a nuisance, doesn't help the situation for those who actually have animals who help them in a medically verifiable way. There's a reason only certain animals get these privileges and we all need to respect those rules.
I bring Kiwi everywhere he's allowed and keep him out of where he's not (which is basically just grocery stores and indoor dining areas and a few businesses that post service animals only). I plan trips around him, we choose restaurants with outdoor dining/pet friendly, I group errands by those he can or cannot come on etc... I just wouldn't feel right lying about him being a therapy animal to bring him somewhere that doesn't allow pets no matter how much I love him and want him with me all the time.