There's absolutely nothing fishy about animal flight companions at all. I do this a lot - for example flying adopted dogs from Soi Dog foundation in Thailand to their new homes in Europe - and it's completely legitimate.
Also, you wouldn't think of how many people relocate intercontinentally and then don't do their homework when it comes to bringing their pets along - thy either booked flights on airlines who don't accept pets, or don't have the required paperwork etc - and then turn to flight companion groups on FB to find someone to take their pet with them.