there is no legal restriction to having someone on the deed who isn’t on the mortgage, but many lenders have it as a policy that the deed much match the mortgage. There used to be a ‘due on sale’ clause that if you transferred the property in any way after the mortgage was filed, then the mortgage would be due, in full, immediately. There are now some federal laws that limit that, and parents/children/spouses, etc can take over the mortgage without penalty.
There are some state laws that make it tricky for married people to own property individually. My mother and brother own her house. My father’s name was NOT on the deed or deed of trust, but he had to sign off on it when they first bought it. My father died recently and it was nice not to have to extract him from the ownership.
You, @tapiocaspider , do need to be aware of the medicaid claw back rules. They can be tricky.