Section 7 of the new Bill is about the right to request to have a pet.
Section 7 (6) of the Bill states, "This section applies to every assured tenancy other than a tenancy of social housing, within the meaning of Part 2 of the Housing and Regeneration Act 2008."
Looking at that 2008 Act, Part 2 relates to Social Housing
Shelter explains Social Housing as being, "provided by either housing associations (not-for-profit organisations that own, let, and manage rented housing) or the local council."
It appears, at this early stage anyway, there will be no automatic right to have a pet for those in Housing Association (HA) accommodation.
It should be noted the Bill refers to the, "Right to request permission to keep a pet" so a single pet and not multiple pets.
As you already have a pet, it appears that even if there were the same right for HA tenants, you would not be able to have a right to request another one.