I hc that most ghosts have an ominous, deathly, oppressive aura to them that typically weirds animals out & scares them. The stronger the ghost, the more strongly it's felt.
Humans usually only feel it on a subconscious level until the higher levels &, to them, every ghost has a slightly different feel to it. For instance, Skulker feels like being in the presence of a predator. (The animal kingdom kind.) While monarchs generally bring to mind being in the presence of a king that could easily have you beheaded at any moment.
Animals just feel a general predatorness.
However, that's just with animals in-general. There are some animals that are almost unaffected by the auras of ghosts unless they specifically register as being a threat, like with Pariah Dark or Dan.
Such animals are generally associated with death or interacts with spirits in one way or another. Things like cats, bats, snakes, & quite a few birds such as cardinals, bluebirds, owls, & vultures, but specifically corvids including but not limited to crows, ravens, & magpies. Several bugs such as worms, spiders, dragonflies, as well as moths & butterflies alike. Not to mention, certain canines such as the xolo dogs of Mexico, the hyenas of Africa, & the jackals of Egypt.
However, tutelaries like Danny have an aura that's slightly different. While there's still the deathly feeling to it, it's more a formidable feeling, like being around someone powerful that you know won't attack you. They give off a sense of comfort & safety to those that ally with them or are being protected by them, but of judgment & coming justice to those that oppose them.
However, there's still a hint of danger surrounding tutelaries that tells people to be wary on instinct. It's almost a contradictory feeling in a way.
For Danny, specifically, it's the feeling of being around a strong & capable soldier that you know would do anything to protect you. Until he bested Pariah, then it slowly turned into that of a strong & courageous soldier crowned king for his good deeds, then after his coronation, that of a benevolent ruler.
As such, animals don't outright run from or attack him, but they are instinctively wary of him.
After the accident, but before the Lunch Lady's first attack, (which was went Danny's Obsession first began to form) animals avoided Danny as he still registered to them as being a predator, but after his Obsession fully formed, they started to register him as being more of a predator who's food source decidedly wasn't them.
However, animals that are around him frequently enough will begin to recognize that he's a protector & will treat him like an alpha & someone to be respected, but it takes a lot of time & association for them to eventually get there. A lot more so than usual & even pets that are generally very friendly by nature tend to be wary of him at first, but not in a "this person is bad" way. More so, a "this is a stranger that might be dangerous" way.
The specific animals associated with death & spirits mentioned above, on the other hand, are able to immediately recognize Danny's protective aura for what it is & instantly see him as being very much a non-threat unless attacked.
Of course, they don't really follow him or try to get his attention or anything unless they recognize him specifically. They initially just recognize him as being a non-threat until they get used to him specifically, then he slowly becomes more & more friend-shaped to them.
Like, butterflies, moths, & dragonflies will just land on him for rest even if he's moving. Corvids, cardinals, & owls will perch on his head or shoulders & just chill. Cats & bats love him though.
For more, go to me full Ghost Zone Masterlist.