How can we evaluate a polynomials in a group instead of a field? (verifiable secret sharing on elliptic curves)
It is correct to me. Polynomial is not evaluated in the elliptic curve group $G$, but in $F_q$ (where $q$ is the order of $G$), which is a field because $q$ is a prime number. 

To be clearer, I add brackets to the equation you gave: $$X_i = \prod_{j=0}^{t-1} (C_j)^{{i}^j} = \prod_{j=0}^{t-1} (g^{\alpha_j})^{ i^j}= (g)^{ \sum^{t-1}_{j=0} \alpha_j i^j}$$ You can see the polynomial is evaluated in the exponent, that why it is in $F_q$.

Maybe it is easier to see if using the Elliptic curve notation (let $P$ be a generator of the EC group $G$, i.e. $g$ in the above equation):

$$X_i = \sum_{j=0}^{t-1} ({i}^j\cdot C_j) = \sum_{j=0}^{t-1} ({i}^j\cdot a_j\cdot P)= (\sum_{j=0}^{t-1} a_j\cdot {i}^j )\cdot P$$