Let $A$ be a not necessarily commutative unital ring with a unique simple module (up to isomorphism). Let $\mathfrak m$ be the annihilator of this simple module, which is a two-sided ideal. We claim that $\mathfrak m$ is a maximal two-sided ideal. If $I$ is a maximal left ideal, then $A/I$ is a simple module and its annihilator is contained in $I$, since any annihilating element must kill $1+I$. If $J$ is a two-sided ideal contained in $I$, then $J$ must annihilate $A/I$, since if $x\in J, y\in A$, then $x(y+I)=xy+xI\subseteq I$, since $xy\in J\subseteq I$. Now, if $M$ is a maximal two-sided ideal (which exists by Zorn's Lemma), then there's a maximal left ideal $I$ containing $M$ (again by Zorn). Then, $R/I$ is simple and its annihilator is a two-sided ideal containing $M$ and thus equal to $M$, which also equals $\mathfrak m$ because there's a unique simple module. Hence, $\mathfrak m$ is the unique maximal two-sided ideal.
If $A$ is an Artinian ring, then $A/\mathfrak m$ is also an Artinian ring (since any infinite descending chain of left ideals in the quotient lifts to an infinite descending chain in $A$). Furthermore, $A/\mathfrak m$ is a simple ring since $\mathfrak m$ is a maximal two-sided ideal, so by Artin-Weddenburn, $A/\mathfrak m$ is isomorphic to a matrix algebra over a division ring. Is this true if we don't assume $A$ is Artinian?
Let $m$ be the annihilator of a simple right $A$-module called $S$.
Then $S$ becomes a simple and faithful $A/m$ module, so that $A/m$ is a right primitive ring. These may or may not be Artinian, and the Artinian ones are precisely the simple Artinian rings (square matrix rings over division rings.)
one isotype of simple module
Now additionally require $A$ to have one isotype of simple right module.
You're right that every maximal right ideal must contain one particular two sided ideal, and it is the unique maximal ideal of $A$. Furthermore, it is the Jacobson radical of $A$.
Additionally, every maximal right ideal is essential in $A$, and the unique simple module is singular and nonprojective.