I have checked with a couple mission scholars to see what they can tell us. The consensus seems to be the missions and settlers probably did introduce both animals. Cats were used on Spanish ships to control rats. For much of the Spanish era, ships from San Blas (on the Baja peninsula) brought supplies and missionaries to California and a few wily cats may have escaped or been taken as pets. There is a story about one of the missionaries (Fr. Uria) who had a collection of cats. Dogs were used to control sheep on the mission lands. All of the missions had large livestock herds. Mission San Luis Rey had the most sheep (26,100 in 1832) but San Jose, San Antonio de Padua, San Gabriel and San Diego also had large herds.
Scholars are pretty careful about making assertions that can’t be backed by written records or other hard evidence. One of the experts I consulted found archaeology reports for the Santa Clara mission site occupied from 1779-1822. The animal bones discovered at that site are: cattle, sheep, pig, cat, skunk, rabbit, small rodent (burrowing into stratum), chicken, rockfish, halibut, and sole.