Well, aside from the usual go-to service pokémon that is togetic and togekiss, swablu and altaria are perhaps the most popular, due to their soothing feathers and equally soothing voices, coupled with their level of comfort and calmness around humans. Altaria especially are good choices, as not only can their wings double as stim toys (this doesn’t hurt them), but also, for those who suffer from anxiety and don’t mind physical contact, an altaria is rather effective in creating a sense of safety for their partners by enveloping them with its wings and humming a calming melody.
The only other pokémon often used for this purpose is jumpluff, whose cotton hands and slow, graceful movements through the air are often said to be rather calming in nature. However, it should be noted that as a pokémon that relies on spore-based techniques (and that stores such spores in its hands), jumpluff may not be effective for those who have allergies.
Beyond that, I admit that the selection for flying-types is rather limited, although anyone can train their pokémon to be companions if they have enough patience and time.
The Pokédex According to Bill 