Let $G = GL_2(k)$ for simplicity where $k$ is an algebarically closed field. When $X \subseteq \mathbb{A}^n$ is an affine variety $k[X] = k[x_1, ..., x_n]/I(X)$ where $I(X) = \{ f\in k[x_1, ..., x_n] : f(P) = 0 \ \forall P \in X \}$. So what does it mean when people use the notation $k[G]$?
In this case is it that $$ k[G] = O_{\mathbb{A}^4}(U) $$ where $U = \{ ad - bc \not = 0 \}$?
If $X = \operatorname{Spec} A$ is an affine scheme over an algebraically closed field $k$, then $k[X]=A$. Now, if $G$ is an affine group scheme, say $G=\operatorname{Spec}B$, then we still have that $k[G] = B$. However, since $G$ is an affine group schemes, the underlying $k$-algebra $B$ must have the structure of a Hopf algebra over $k$.
Here are some examples:
$$ \operatorname{GL}(1,k) = \operatorname{Spec} k[x,1/x], \ \ \ \ k[\operatorname{GL}(1,k)] = k[x, 1/x]$$
$$ \operatorname{GL}(2,k) = \operatorname{Spec} k[x,y,w,z,1/(xz-yw))], \ \ \ k[\operatorname{GL}(2,k)] = k[x,y,w,z,1/(xz-yw))]$$ $$\operatorname{SL}(2,k)= \operatorname{Spec} k[x,y,w,z]/(xz-yw), \ \ \ k[\operatorname{SL}(2,k)]=k[x,y,w,z]/(xz-yw-1) $$