As with Airbnb everywhere, whether a Swedish listing welcomes pets is the individual host's decision, stated in the listing's House Rules section. Hosts with a standalone house, cabin or stuga often welcome pets more readily than hosts in shared apartment buildings, where house rules set by the building itself can rule pets out regardless of what the individual host would prefer.
If the listing is silent on pets, message the host directly before booking rather than assuming. Swedish hosts can decline or cancel a booking if an undisclosed pet turns up, and Airbnb's standard cancellation terms rarely favour the guest in that situation.
Worth asking about specifically: any pet cleaning fee, whether there's a securely fenced outdoor space (useful given allemansrätten's seasonal leash rule covered in Activities), and how close the property is to a lake, forest trail or other dog-friendly outdoor spot.

