To be accurate, that’s only stipulated in advertising. When it comes to a tenancy agreement, typically you have a clause that states that the LL should not “unreasonably deny” a request for a T to have a pet. Tracy has just posted an example of a situation that would be a reasonable way to deny that request: the pet caused damage.
But if a T said their son wanted a small lizard in a tank in his bedroom, a LL would be hard pushed to find a reasonable way to deny that request.
For me, denying a request that is reasonable simply sours the relationship with the T and marks the beginning of the end. It’s my property but it’s their home and I need to bear that in mind whenever I get a request.