One limit stands behind both properties
Rather than raising the liability limit on a homeowners policy and again on a landlord policy, a single umbrella sits above both and responds to whichever produces a claim. It is almost always cheaper than lifting each underlying limit to the same height, and it is simpler to manage.
Every property has to be listed
An umbrella covers the exposures scheduled on it. A rental the insurer was never told about is generally outside the cover, and that is the single most common failure on a household umbrella. Adding a property takes a phone call.
The rental is the likelier claim
Tenants, their guests and a property you are not watching daily produce more liability than the home you live in. For a household with one rental, that property is usually the reason the umbrella is worth having at all.