Maximillian is the grand, extended spelling of Maximilian, itself already a name of maximum ambition from the Roman Maximus (greatest). With an extra L and four full syllables, this is a name for owners who want their pet to be, at minimum, nominally the greatest thing in the room. The humor is self-aware, and it works.
The Over-the-Top Name Aesthetic
Maximillian on a pet is a deliberate escalation. Great Danes and Bernese Mountain Dogs can actually carry the name without irony. Any dog of imposing physical presence earns it without the joke landing.
Nickname Ecosystem
Maximillian's real value is its nickname depth: Max, Maxi, or the full ceremony of "Maximillian, come here at once." The formal name provides gravitas; the daily nickname provides practicality. See Max for the widely popular short form, currently one of the top male dog names nationally.
Counter-Reading: The Double-L Is a Spelling Trap
Standard spelling is Maximilian with one L. The double-L version reads as intentional distinction or spelling error depending on who's reading it. Your vet's records will probably standardize it. If not, Max is simpler, equally beloved, and requires no explanation to anyone anywhere.