I think part of my issue here may be that I don't know the correct terminology to find an example, as everything that I've attempted to search has been unhelpful.
I'm trying to come up with an example of a monoid where $ac=bc$, but $a\neq b$. My understanding is that for this to be true, $c$ cannot have an inverse, but I'm having trouble coming up with an actual example where this is true.
You might be looking for the terminology "cancellation property." Possibly interesting: if a commutative monoid has the cancellation property, then the Grothendieck construction gives a group the monoid embeds in.
A commutative monoid without the cancellation property is $M=\{0,1\}$ with $0+x=x$ and $1+1=1$. (I believe I learned this from Mathematics Made Difficult by Linderholm.)
A noncommutative monoid example is $n\times n$ square matrices. It is not hard to find three matrices $a,b,c$ with $a\neq b$ and $ac=bc$, even without $c=0$.