As per my understanding:
Functions:
$\sup_{x \in I} f(x) = \sup \{y \in \mathbb{R}: y = f(x), x \in I\}.$
Example: $f(x) = x^2$ and $I = [0,3)$:
$\sup_{x \in I} f(x) = 9$.
Sequence of functions:
$g(x)=\sup_{n \in \mathbb{N}} f_n(x) = \sup \{ y \in \mathbb{R}: y = f_{n}(x), n \in \mathbb{N}\} \tag{1},$ where $x$ is fixed.
Example: $f_n(x) = x^n$
$\forall x> 0, x \neq 1, g(x)=\sup\{x,x^2,x^3,x^4,x^5,......\}= +\infty$.
$x=1: g(x)=\sup\{1,1,1,1,....\}=1$
$x=0: g(x)=\sup\{0,0,0,0,....\}=0.$
$x = -1$: $g(x)=\sup\{-1,1,-1,1,....\}=1$
$\forall x<0, x \neq -1$: $g(x)=\sup\{x,x^2,x^3,x^4,x^5,.....\}=+\infty$.
But there is also another way of taking the supremum of a sequence of functions:
$\sup_{x \in I} f_n(x) = \sup \{y \in \mathbb{R}: y = f_n(x) , x \in I\}=(g_n)_{n\in \mathbb{N}} \tag{2}$
Example: $I =[0,3)$: $ (g_n)_{n\in \mathbb{N}} =\sup_{x \in I=[0,3)}f_n(x)=3^n.$
My conclusion: $(1)$ and $(2)$ are two different ways of taking $\sup f_n(x)$ and are used according to the specific needs.