We know the Hopf link owns the name of $2^2_1$ for Alexander–Briggs notations. (And there is another two component links is $4^2_1$.)
I learned that "$4^3_1$ is not usually written as any three component link with 4 crossings (and no un-linked components) must be the connect sum of two Hopf links." from this post.
question: So how do we call this following three component links in Alexander–Briggs notations below? Could it be $4^3_1$? Or what ever other notations we use to describe this links? Is this connect sum of two Hopf links? Can we write it as $2^2_1 \oplus 2^2_1$? So if there are sum of N Hopf links, can we denote $$\bigoplus_N 2^2_1 \;\;\;\;?$$
You have to be careful when taking the connect sum of links as the operation is not well defined until you have specified which components are being connected together (we also need to consider orientation) For instance if we have three Hopf links $H_1, H_2, H_3$ with components $C_i^1,C_i^2$ of $H_i$, then $(H_1\#_{C_1^2,C_2^1}H_2)\#_{C_2^1,C_3^1}H_3$ and $(H_1\#_{C_1^2,C_2^1}H_2)\#_{C_2^2,C_3^1}H_3$ are not the same link$^*$ because the first can be disconnected into a trivial link with three components by removing a single component (the 'central' component $C_1^2$), whereas the second can not be similarly disconnected.
Note, the $\#_{C_i^j,C_{k}^{l}}$ notation I've used above is just my own notation - I don't know if there is a standard notation which means 'take the connect sum along the component $C_i^j$ of the left link and the component $C_k^l$ of the right link'.
It turns out that the connect sum of two Hopf links is invariant under a choice of components to connect (or even orientation of components) for pretty obvious diagramatic reasons so you can write $H_1\#H_2$ without confusion, as long as you don't then plan to take the connect sum of this with other links. As such, I think you would be fine in calling the link you have presented $2_1^2\#2_1^2$ - I don't know if this is standard notation and I am not an authoritative source on knot theory.
$^*$ Note the second connect sum is done on different components of the link $H_1\#_{C_1^2,C_2^1}H_2$ at $C_2^1$ and $C_2^2$ respectively.