Can I just pick up a pigeon and keep it?

You cannot just pick up a pigeon and keep it as a pet without considering legal, ethical, and practical issues, though laws vary by location and type of pigeon. Feral pigeons in cities are descendants of domesticated rock pigeons and are generally not protected by federal migratory bird laws in the US, so in many places it is not illegal to catch a feral pigeon, but it is not recommended because feral pigeons can carry diseases like histoplasmosis, cryptococcosis, salmonellosis, and parasites like fleas, ticks, and mites that can affect humans, and they are wild animals not used to handling and may be stressed and bite. Additionally, many cities have local animal cruelty laws that prohibit capturing wild birds in inhumane ways. If the pigeon is a banded homing pigeon or pet pigeon that is lost, it likely belongs to someone and you should try to find its owner by reading the band. If you want to keep pigeons, the better approach is to adopt or buy domesticated pigeons from breeders or rescue organizations that sell healthy, tame birds bred for captivity, and provide proper housing like a loft, coop, or large cage, with food like seed mix, peas, and corn, water, and veterinary care. Keeping a single pigeon alone is not ideal as they are social and need companionship of other pigeons or daily interaction. So while you may legally be able to catch a feral pigeon in some places, it is not advisable, and you should adopt domesticated pigeons instead.