Donie, Doney may have multiple origins. In addition to the donnas etc. derived from the Latin dominus, domina, there are other possibilities for a particular occurrence of the name.

Donie is not uncommon, both as a first, and last name. It appears as a first name for both males and females. Baseball nuts know of Donie Bush who played up to the time of WW1 or later. Donie is a variant of Don, or taken from the rather common last name. The city of Donie, TX, received its name in 1898; no antecedent, just a mistake for Douie.

Doney also in not uncommon as a first and last name. One branch is traced to Cornwall, 16th century. Variants Dounay, Dony, Downie, etc.
The Doney Trail in Arizona, Doney Lake, MT and Doney Park, CA are named after people, not animals.
The name de Dunnay goes back to 13th century Scotland. Some descendents are Doneys.
Dony also appears as a first and last name.

In Spain, the picturesque old town of Doney and environs has been much photographed. Some of its people emmigrated to Mexico-New Mexico in the 17th century. Doney also is an old English name for a thicket or enclosure. Neither of these seem to have any bearing on the usage sought here.