The Generator point and Mod P in ECDSA
The video (at least, where the question links) illustrates the Baby-step/Giant-step Discrete Logarithm method in the multiplicative group modulo $p$, for prime $p$. That is the set $\{1,2,\ldots,p-2,p-1\}$, under the internal operation multiplication modulo $p$.
This group has essentially nothing to do with an Elliptic Curve group. The principle of Baby-step/Giant-step is the same regardless of group, however the operation analog to multiplication modulo $p$ is extremely different in an Elliptic Curve group.

It is important for security that the order of the group generated by $g$ (that is, the number of distinct values $g^x\bmod p$ can take) is prime, or at least has a large prime factor. When working in the multiplicative group modulo $p$, the order of the full group is $p-1$, and the order of $g$ is some divisor of that. One option is to choose $p$ such that $(p-1)/2$ is prime ($p$ is a so-called safe prime). Now we can take any $g$ except $g=1$ and $g=p-1$, and it's order will be $p-1$ or $(p-1)/2$, which both have a large prime factor $(p-1)/2$.
There is no imperious reason to choose a generator $g$ that is prime. It happens that $2$ and $3$ are popular, and prime; and some people have found satisfactory to choose $g$ prime.
There's even less reason to worry about prime $g$ in ECC groups. In secp256k1, both the $x$ and $y$ coordinates of the generator point are large even integers, thus not prime. What matters is the order of the group generated by $g$. That is noted $n$, and the curve parameters are usually chosen such that $n$ is prime.