Cross species transmission is rare and while some cases have been reported, they're really the exception rather than the norm.
If there was really that big of a risk where we'd need mammals to isolate as well, then veterinarian services the world over would have collapsed under the weight of house pets contracting the virus.
edit: curious about the downvotes. Perhaps you have proof that has eluded the virologists researching SARS-CoV-2. Please do speak up ;)
Why would they get severe disease when they get infected? Even most humans don't, let alone other species with different receptor setups. The virus is absolutely in non-human animal populations almost everywhere, and is not a candidate for eradication.
If there was really that big of a risk where we'd need mammals to isolate as well, then veterinarian services the world over would have collapsed under the weight of house pets contracting the virus.
edit: curious about the downvotes. Perhaps you have proof that has eluded the virologists researching SARS-CoV-2. Please do speak up ;)