>>35125993Snowy has the Modest nature (according to the event one in the games), so like her trainer she is very polite and reserved. Even before she associated with Lillie, Snowy has been very reserved when it was around other Pokemon (compared to the more energetic and outgoing Samson Oak's Vulpix). Snowy isn't shy or timid or anything; she gets along with other Pokemon, but she doesn't express herself all too much unless she trusts those she is with.
In many ways, Snowy has many of the character traits of her trainer, which is why she chose Lillie. As an aside, the anime even references Snowy's nature (Modest) when they showed her disliking the Akalan curry in SM035 since it is spicy. Pokemon with -Attack natures (such as Modest) hate spicy food in the games.