To start with the big question, the flute can be played in both ways, either sitting or standing. For example, practicing on your own at home or just by yourself you can choose to either sit or stand. Typically it doesn't matter all that much when it comes to individual playing. However, during band, lessons performances, concerts, etc. You and your band mates will most always be sitting when performing. For the lads and ladies with heavier instruments sitting is a lifesaver. As well as it would be a bit peculiar to watch a concert performance with the entire ensemble standing. Percussion and rhythm section typically stand to preform and practice because of the instrument itself.
More specifically on flute, when you sit there is a "correct" way to sit to allow for full breathing and a professional look. This is achieved by sitting on the chair with both feet on the ground not spread apart, not crossing tall broad posture, but not so much it is unnatural. But your posture when you sit plays a key role in how well your breathing will be when you play. Slouching is a big no no when it comes to sitting and playing. Standing and playing flute is typically done with lessons (instructed by the teacher) individual practice if you so choose and for solos. When you stand whilst playing flute make sure to have your stand be about shoulder width and knees slightly bent. Depending if you are soloing or not, make sure to look in front of you or even maybe a little upwards if you are looking at a target that is elevated from the ground. Lastly, always, always always, when sitting or standing make sure your flute is parallel to the ground and not tipping one angle.