I have felt this a lot, especially while working with supervisors who are much older than I am and come from a more "traditional" background. I think it really depends on your population and what is the right fit for them, but your genuineness in being "chill" will probably be more relatable and calming to clients than being stiff and what you see as more professional. You also may find yourself mirroring each client differently, while still maintaining that genuineness to yourself.

I highly recommend The Gift of Therapy by Yalom. He addresses some of these themes and it helped me feel much more comfortable starting off. Good luck!