What purpose does this broken plastic clip serve in the MacBook Pro?

That clip holds the bottom piece snug in the middle. I would remove the broken plastic to avoid bending the metal tab that matches this on the bottom case. Also, you don't want it pushing on the connectors that are delicate and behind the metal cage above it. Lastly, if the piece breaks free it could wedge in the wrong spot (like a blower or worse) and cause more harm than it should.

You can see about a repair part from Apple by contacting AppleCare and this sort of thing is often sold used when a machine is liquid damaged or otherwise scrapped for parts. Many places sell used parts like ebay, amazon and ifixit.

THIS PART IS IMPORTANT!!!

Improper holding of bottom cover can change internal airflow completely (cold air will flow at the bottom not washing motherboard). Especially in my MacBook Pro 17 2010... botom cover is significantly protuberant. So this clip/holder have good sense.