why is a regular open (closed) set called "regular"? Certainly it doesn't have to have an obvious reason other than being sort of "simple" or "nice", but its resemblance to the "regular" in "regular semigroup" (https://en.wikipedia.org/wiki/Regular_semigroup) makes me wonder if they are indeed related.
I know that these two "regulars" are actually on different types of objects: the regular open set is certain type of set, while a regular element in a semigroup is an element $a$ having some algebraic property, i.e. $\exists x : axa = a$, but if we look at the closure/interior operation underneath the definition of a regular open/closed set (which are like algebraic operations as well), they are probably similar.
But how?
 A: General note
The adjective "regular," along with the adjective "normal," is probably one of the most overloaded adjectives in all of mathematics. Whenever mathematicians find a condition that makes things nice they are apt to try to find a good adjective to package it up, and early on especially, they didn't cast around very long: they had better things to do, like write groundbreaking math.
For those of us working in the second half of the 20th century and in the 21st century and beyond, we have to work a little harder while naming :)
Let me make a few general comments about the two notions that will set the stage for a comparison.  I will make the case that there is no close relationship between these two particular terms.
The set operations

but if we look at the closure/interior operation underneath the definition of a regular open/closed set (which are like algebraic operations as well), they are probably similar.

This here is related to the Kuratowski 14-set problem. If $k$ is the closure operator and $c$ is the complement operator on a topological space $X$, then the monoid generated by these two operations together has at most 14 distinct elements. For some spaces it will be smaller.
Now, if you alias the interior operator as $i=ckc$, you get a submonoid of that monoid with at most $7$ elements: $\{I, k, i, ki, ik, iki, kik\}$ where $I$ is the identity operator.  This is because the nontrivial kuratowski relation yields both $ikik=ik$ and $kiki=ki$ Again, for some spaces it will be smaller.
By definition, a subset $S$ is regular open if $S=ik(S)$, so it is a special class of subsets of $X$.
(von Neumann) regularity of semigroup elements
I say "von Neumann" regularity because even in ring theory the adjective "regular" is overloaded, and this is what we use to help distinguish this particular flavor.
I don't know if I can do justice to the notion of "regular element" of a semigroup in general. My main experience with them is through rings, but I suppose they are better known by semigroup theorists.
When for an $a$ there exists an $x$ such that $axa=a$, we can also say that "$a$ has an inner inverse" or sometimes "$x$ is a weak inverse of $a$."  Even though $ax$ and $xa$ aren't necessarily the identity, but they are at least idempotent, and you can "cancel" $x$ off of the left side of $a$ by using $a$ on the left, and so on.
Comparison
Now, you can already see that the semigroup notion is far departed from the set notion: the former discusses sets defined by a condition dictated by the topology, and the latter discusses the weak-invertibility of a binary operation. There is no  concept directly bridging the two.
What you may be picking up on is the relation of the latter to the former's operator monoid.  Indeed, in that monoid every element is idempotent and hence regular.
