I was very suprised when I found this post since I am not aware of this post although the Clifford algebra is my Phd subject. Indeed, the construction of Clifford algebras can be regarded as a left adjoint on a different category $\mathrm{HPL}^d(S)$, which also work for any forms of higher degrees with non-commutative coefficients. You can find this adjunction in Section 3.1 of my thesis https://drive.google.com/file/d/1MKS6Y1V1lEtt60C9ZARAD-UyVlxtr8X3/view?usp=drivesdk
Reading through the answers, I think the problem is that the notion of forms relies to much on a concrete presentation, as well as the coefficients of form is restricted to the commutative base ring.