In usual mathematical notation, commas do not have any specific fixed meaning — they just serve to separate things in a list of things.
Events are, formally, subsets of the sample space $\Omega$. While there are several ways of defining events, in your example the events $E_1$ and $E_2$ are literally written as sets by enumerating their members (i.e. the primitive outcomes that are included in the events) inside a pair of braces.
In this enumerative set notation, the commas simply separate the members of the set, so that you can e.g. tell the difference between the sets $\{1,2\}$ and $\{12\}$.
Now, since a set is the union of the singleton sets (i.e. the elementary events) containing each of its members, you could certainly interpret the set $\{1,2\}$ as $\{1\} \cup \{2\}$. And since the members of a sets are always distinct, and their singleton sets thus always disjoint, you could also replace the union $\{1\} \cup \{2\}$ with the symmetric set difference $\{1\} \mathbin\triangle \{2\}$ (which is just the set-theory version of XOR, just like $\cup$ is the set-theory version of OR). But that's really overthinking it, in my opinion — fundamentally, $\{1,2\}$ is just a set containing a list of members, and that's how you should think about it.
As for notation like $P(E_1, E_2)$, what's inside the parentheses is just a list of events, separated by commas to make it clear that we're not talking about some single event named $E_1E_2$ or whatever. With events represented by single-letter variables, that's arguably not really necessary, but it becomes useful when we have more complicated events like $X = Y$ or $A \setminus B$.
Now, by common convention, a list of events is interpreted as the intersection (AND) of those events, such that $P(E_1,E_2) = P(E_1 \cap E_2)$ or, using logical connectives instead of set-theory ones, $P(E_1 \land E_2)$. However, that convention is by no means universal, so if you want to be sure to avoid ambiguity, you should explicitly use $\cap$ (or $\land$) to denote the intersection of events.
However, I do personally confess to often being lazy and just using comma-separated lists of events, despite the risk of confusing people unfamiliar with the convention, simply because a comma is quicker to type and takes less space than $\cap$. (Besides, not everyone is familiar with set operators or logical connective symbols either, so when I do explicitly include the connectives while writing to a general audience, I often spell them out as in $P(E_1 \text{ and } E_2)$.)