Disneyland only allows "trained service animals" (which they define) into the park. (The link doesn't explicitly disallow ESA animals but Google "Disneyland service dog" and the second link is a question/answer format which says they are not allowed.)

https://disneyland.disney.go.com/guest-services/service-animals/

Google "Great Dane service dog" and the first two results are about how they are used for adults with balance and mobility issues:



https://integrativevetmedcenter.com/blog/why-great-danes-make-good-mobility-service-dogs/#:~:text=Great%20Dane%20Service%20Dogs%20provide,of%20life%20for%20their%20owners.

The upshot is that, if the dog wasn't a true service dog, the owner at least answered the magic ADA questions as if it was. Also, the breed is used as a service dog based on the combination of its size and relative docility. The lady could have been less abrupt about it, especially explaining it to a little kid, but unless she lied to get the dog in, she wasn't wrong.