I've clicked through about a million questions on here and and on CS Stack Exchange and not yet come across an unequivocal "yes" or "no" to the following:
Is the empty string the same thing as the empty set?
Sipser ("Theory of Computation") defines a string as any tuple with elements from a given alphabet (the alphabet can be any nonempty finite set). As far as I can tell, it follows that this question is equivalent to "is the zero tuple the same as the empty set"? I am not sure about this in turn because the set-theoretic definition of tuple has some nested sets, so I could at least imagine the zero tuple being defined as the set containing the empty set or something like that.
As JMoravitz describes, the short answer is it can, but it doesn't matter as long as your definition of empty string has the expected properties, such as being the identity under concatenation. If it's desirable for a string or tuple to potentially be non-finite and have trivial projection functions, as is common in computer science, then a convenient definition is that a $k$-tuple over $S$ be a function $\mathbb{N}_{<k}\to S$. In this case indeed the empty string is the trivial function from $\varnothing$, which has the usual encoding of $\varnothing$ as a set. Concatenation can be defined on these functions with some care.
But! To drive the point home, this isn't a typical set-theoretic definition of $k$-tuple, where one might instead usually use something like iterations of $A\times B\mapsto\{A, \{A, B\}\}$. In this case the empty set isn't an identity, so it can't be the empty string. It's instead a 1-tuple with element $\varnothing$! The notion of 0-tuple is not well defined here. This doesn't mean you can't make strings out of this notion, you just need a special symbol for the empty string (say, $S\cup\{S\}$ if $S$ is your symbol set) and to adjust your notions of concatenation, etc., appropriately.