The Answer
Pluto the Planet – discovered February 18, 1930, named March 13, 1930
Pluto the Dog – first appeared in 1930, but not under the name Pluto until May 8, 1931.
The (Dwarf) Planet
Pluto the planet (at the time) was discovered by astronomer Clyde W. Tombaugh on February 18, 1930. The discovery and the name Pluto was announced on March 13, 1930. The planet was named by 11-year-old Venetia Burney after the Roman god of the underworld when her grandfather passed along her suggestion to his Oxford University acquaintance, Clyde Tombaugh.
Mickey Mouse’s Dog
Mickey Mouse’s pet dog first appeared in a cartoon called The Chain Gang in 1930 but was unnamed at the time. He then appeared as Minnie Mouse’s dog in The Picnic but under the name Rover. The first appearance of the dog named Pluto was in The Moose Hunt in 1931.
It’s unclear (to this author, at least) whether the recent naming of the planet had any influence on Walt Disney’s selection of Pluto as his character’s name.