Does state or federal government offer financial aid to build an accessory dwelling unit for an elderly person to house their care aide?

I am not familiar with any type of State/Federal aid for such stand alone dwellings. You would need to contact the State offices to see what programs, if any, they would have. You probably could receive a standard mortgage as said dwelling would be on the low end of cost.

I think one major issue would be zoning, if said property has enough property distance from neighbors to erect something the size of a double car garage, and to use for occupancy.