Why is fluid leaking out of my belly button?

Sounds like serous body fluid (this is normal body fluid) is oozing out.  It should stop shortly but in the mean time let your PS know about this.  As your body heals, it will re-establish it's ability to drain the fluid from the surgical area but until then it can collect into a seroma or it drains out (better option).  So while it may be messy, it's better than letting it collect under the skin.

Thank you for your inquiry. I would see your plastic surgeon ASAP, you may have a seroma, fluid collection, under the skin that is draining through the belly button. It is important to discuss how to treat and in particularly how to reduce the chance of an infection.

The belly button can often take longer to heal because of decreases in circulation to its stalk during surgery. Fatty tissue around the belly button can similarly loose some blood supply, become necrotic, or liquefy, causing drainage. Another reason could be a deeper suture in the area that is inflamed. I would recommend an exam fairly prompty in order to treat the condition before it worsens or gets infected.