Is there a toy example of an axiomatically defined system/ structure?

This may go beyond what you are asking for, but there is a wonderful book called Introduction to the Foundations of Mathematics by Raymond L. Wilder. I provided its axioms and an example of how they could be satisfied using SET cards in MESE 2528.

Here is again a list of the axioms:

The full book is available online (open access) and I have derived quite some joy over the years from proving the various theorems that follow the above-mentioned axioms. Wilder himself begins his discussion of these five axioms by showing that they can be "satisfied" with an interpretation around people as members of clubs, and remarks that it is achievable with just four people; so, he continues, this cannot be a full description of (e.g.) Euclidean geometry, since we know there are many more than four points in the plane!

The theorems that Wilder uses begin as innocuous; according to my own notes, the first one is:

Theorem 1 Every point is on two distinct lines

and become increasingly more difficult; e.g., given: $n$ total points; let $m$ be the number of lines containing a given point - this has already been shown as well-defined in an earlier theorem; and let $r$ be the number of points on a line - also shown well-defined in an earlier theorem. Then one has:

Theorem 10 The total number of lines is $m \cdot r$.

The book is from 1952 (the link above goes to its reprinted second edition from 1965) but I think it holds up quite well. Even if the full book turns out not to be ideal for your present purposes, I would recommend this as a reference; there are some nice problems and explanations scattered throughout.

Howard Eves has a nice example in Great Moments in Mathematics (After 1650), Lecture 36: a set $K$ of elements and a relation $R$, with the following 4 postulates:

1. If $a \ne b$, then either $aRb$ or $bRa$.
2. If $aRb$, then $a \ne b$.
3. If $aRb$ and $bRc$, then $aRc$.
4. $K$ consists of exactly four distinct elements.

He then proves seven specific theorems (with more in the exercise list), and also discusses changes if some postulates are altered. Then he provides applications of those theorems in a variety of fields (arithmetic, geometry, and genealogy).

This approach below does not work equally well in all countries, but examples based on changing the rules of a familiar sport/game I think can help some students with regard to ideas related to axiomatics. In this column (basically about the taxicab plane) that I wrote some years ago for the Feature Column of the American Mathematical Society I have a brief discussion of this idea in the context of American baseball.

Scroll down to the section on "axomatics."

http://www.ams.org/samplings/feature-column/fcarc-taxi

Usual baseball - (3,4,9)-baseball - has 3 strikes for an out, 4 balls for a walk, and 9 innings. What happens when one changes these parameters?

One example from Euclidean geometry in planes. Consider the following three axioms:

1) There exist three non-collinear and pairwise different points in the plane.

2) Every 2 distinct points can be connected by a unique line.

3) For every line $g$ in the plane and every point $A$ not on $g$ there exists a unique line $h$ through $A$ (i.e. $A \in h$) with $g \cap h =\emptyset$.

If you assume these axioms to be true, then you can easily prove that the plane consists of at least 4 points in the plane:

Suppose you have a plane. Axiom 1) yields that there are at least 3 non-collinear pairwise different points $A$, $B$ and $C$. According to axiom 2) there is a unique line $g$ through $A$ and $B$. Yet, $C \not \in g$ since $A$, $B$ and $C$ are not collinear due to 1). Now axiom 3) tells you that there is a line $h$ through $C$ with $g \cap h =\emptyset$. According to 2) $h$ is a unique line through (at least) 2 points with one being $C$ by construction. Hence, there has to be a fourth point $D$ in the plane with $D \in h$.

I hope this example helps a bit. I tried to keep it simple and did not use any definitions for a line for example. In a more scientific context this should, however, be done. Since you asked for a toy example the above example should be sufficient.

For my money the best possible "toy" examples are the MU-System and the pq-System introduced and defined in Chapters 1 and 2 of Douglas Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid. The latter, in particular, is extremely useful for stressing the delicate balance between syntax and semantics: on the one hand, having an interpretation of the words and axioms in your system can play an important role in helping anticipate which theorems are likely to be true and which ones false; on the other hand, one's reasoning should not depend on that interpretation, because there may be other interpretations that are equally valid.

I like this example better than some of the geometric ones people are suggesting because it's hard for beginners to suppress their intuition about what the words "point", "line" and "plane" mean. (Hilbert's apocryphal quote about "tables, chairs and beer mugs" fits in well here.) In Hofstadter's pq-System, the reader genuinely doesn't know what p and q mean (or might mean) until after they have figured out some of the theorems and non-theorems of the system.