Confusion over spin representation and coordinate ring of orthogonal Grassmannian

Let me first treat the case $n = 1$. Then $G\cong PSL(2,\mathbb C),P\cong (GL(1,\mathbb C)/\mathbb Z/2)\ltimes \mathbb C$ embedded as upper triangular matrices, $G/P\cong \mathbb{CP^1}$, and $\omega_1$ defines the line bundle $O(1)$ over $\mathbb{CP}^1$. The root of your confusion is that this line bundle is not $G$-equivariant; however, after passing to the covers $\widetilde G = SL(2,\mathbb C),\widetilde P = GL(1,\mathbb C)$, it is $\widetilde G$-equivariant since it is the associated bundle construction $\widetilde G\times_{\widetilde P,\omega_1}\mathbb C$ on which $\widetilde G$ acts from the left. And indeed, $V^{\omega_1}$ is the 2-dimensional representation of $\widetilde G$, which is not a $G$-representation.

In the general case, we have $P = GL(n,\mathbb C)\ltimes \mathbb \{A\in \mathbb C ^{n\times n}\mid A^T = A\}$. The weight $\omega_1$ does not define a character of $P$, but of a double cover $\widetilde P = ML(n,\mathbb C)\ltimes \mathbb \{A\in \mathbb C ^{n\times n}\mid A^T = A\}$, where $ML(n,\mathbb C) = \{U\in GL(n,\mathbb C),z\in \mathbb C^*\mid z^2 = \det U\}$ is the metalinear group (the character is $(U,z)\mapsto z$ or $z^{-1}$). This means, again, that the line bundle $L^{\omega_1}$ and its holomorphic sections $V^{\omega_1}$ only carry representations of the universal cover $Spin(2n+1,\mathbb C)$. This works just as well for the line bundles $L^{k\omega_1}$ whose holomorphic sections define the higher degree components of the coordinate ring: The action of $G$ is well-defined precisely if $k$ is even.

More generally, the action of a group $G$ on a projective variety $X$ only gives rise to a projective representation of $G$ on the (linear part of the) coordinate ring of $X$, i.e. a representation of a central extension, with the kernel of the extension acting by scalars in each degree. In this case, the central extension is precisely the spin group.