I don't think that those are the same position.
Let's update our understanding and use other more meaningful categories that better reflect people's lived experiences is a good idea. Let's confine our understanding and hold people in rigid categories that often do not match their lived experiences is not.
John Oliver also has a good segment on this topic, if you're interested.
Also, one could listen to someone such as Erin in the morning to understand the context of the anti-trans sports campaign.
Some of what Erin describes here is that much of the current anti-trans efforts are being funded and pushed by many of the same religious fundamentalist groups that previously pushed "defense of marriage" campaigns and and legislation against gay people.
The market research that these groups have used since losing that debate have shown them that religious arguments against inclusion are generally unpopular. So now they've made a very deliberate, and rather successful, effort to repackage their agenda through the sports topic instead.
As others have described in this thread, "emotional support animals" and "service animals" are different entities with different requirements and privileges.
Service animals generally have specific training to perform specific tasks to support an individual who has a disability. Service animals have fairly broad societal privileges based on the protections described by the Americans with Disabilities Act. You could read more about that here: https://www.ada.gov/topics/service-animals/.
An emotional support animal is a much broader definition which describes an animal whose presence may reduce an individual's distress. Emotional support animals have fewer societal privileges. Some organizations or businesses may allow the presence of an emotional support animal, but others may not. The main legal protection relating specifically to emotional support animals is that the Fair Housing Act protects owners of emotional support animals from discrimination in housing, such as being denied housing or having to pay additional fees.
Another place where you could read more about the distinctions between the two would be: https://www.akc.org/expert-advice/training/service-dog-training-101/
There aren't really any organization that I know of which formally designate animals as emotional support animals, other than some online websites where you can buy and print your own certificate if you are interested in doing so.
There are many instances where a healthcare worker, such as a doctor or therapist, may recommend the presence of an emotional support animal, however. This kind of recommendation is typically based on the healthcare conditions experienced by the human, which the healthcare worker can attest to as being consistent with medical necessity, such that the presence of an emotional support animal is recommended to support the healthcare needs of the human.
NABDad describes a good example of an appropriate recommendation for the use of an emotional support animal in this thread. If you read his description carefully, he says that the doctor made the recommendation based on his daughter's healthcare needs, and then they went and found the guinea pig to serve as a form of healthcare treatment after that. The doctor's assessment was based on NABDad's daughter, not on the guinea pig.
If you would like Brownie to be recognized as an emotional support animal for your roommate, the process would generally be for your roommate to contact a healthcare worker, such as a doctor or therapist, so that your roommate can discuss their healthcare needs.