The short version is: No. Not all systems are identical, but they also aren't random. There are real patterns, and it has been studied A LOT. It's a field of science called Galactic Chemical Evolution, and it's fucking rad.

The long version. No, again :) When the big bang happened, it scattered everything that ever was, but everything at that time was just hydrogen and helium, which is boring. So, at the very beginning, everything was homogeneous because there wasn't much of anything to spice it up.

However, then the stars came along after being formed by those gasses. When hydrogen and helium got cooked, they fused into heavier elements, which is where all the fun stuff got made. Then, the stars would die, and you'd get a supernova. These would spread all the cool elements around, and that's where we get everything we are used to looking at, like planets and asteroids and moons. Everything is made of star stuff.

So, for your questions:

1. Is our system similar to others? Yes, in the big picture. Almost everything everywhere is overwhelmingly hydrogen and helium (98% of all matter by mass). The cool stuff exists in the tiny sliver that's left.

2. Are we more similar to our neighbors? Yep, and that's a great question. Stars that formed us formed others that were once our neighbors, but stars drift over billions of years, so we can't just look out a window and see our neighbor, we have to use chemistry to figure out where we came from. An example is that our current neighbor is Alpha Centauri, but we aren't siblings to them. We're just close to them now, after billions of years.

3. Is it random? No. The stars at the center of a galaxy tend to be more metal rich because more stars lived and died there, while the stars at the outside are lower in metals. Age plays a role as well, because older stars formed before all the cool stuff, and so they had less cool stuff to play with.

Hope that helps. Science is fucking awesome. If you want to know more, look into Galactic Chemical Evolution, especially on YouTube where you can find a lot of educators that explain this with much more depth than I can. PBS SpaceTime is one of my favorites.

From my studies, I remember that solar systems have some kind of identity card based on the % of atoms and isotopes (versions of an atom with more or less neutrons).

The bigger the atoms the more exceptionally energetic the process that created them needs to be: normal star, big star, supernova, neutron star merger... Also some specific isotopes can be created by some processes and not others. https://en.wikipedia.org/wiki/Nucleosynthesis

So one part of the universe where a solar system formed may not have seen the same atoms creation processes as another, so their detailed compositions are different.
Cosmo chemists analyze the compositions of meteorites that formed at different times to try to understand the history of our solar system. For exemple, they may find that this isotope could not have been produced by our sun's birth, and it must have come from a supernova. https://www.quantamagazine.org/what-crystals-older-than-the-sun-reveal-about-the-start-of-the-solar-system-20260302/

They make our origin much cooler by looking at space pebbles.

This is a difficult thing to accurately assess. While we can use spectrometry to find chemical compositions of stars and nebulae and planet atmospheres/surfaces, we cant even be certain about the interior of planets and moons within our own solar system, much less the galaxy.

We can make educated guesses based on the mass/volume and the stability of elements but I don't think it’s an exact science.

However, i would imagine the answer to your question is No, the galaxy is not homogeneous. My reasoning is that near the center of our galaxy, stars are much closer together. This not only makes stellar collisions more likely to occur, but also means there will be significantly more cosmic rays in that space than out here near the rim.

Both cosmic rays and stellar collisions are the (theoretical) primary sources of certain elements. Berylium and Boron are created by cosmic rays fission, and as such id expect their concentrations to be higher in denser star clusters. And I’d expect higher concentrations of heavier elements due to both increased collisions AND due to the decay chains of unstable isotopes created by cosmic ray fission.

Cosmic rays are responsible for producing common isotopes like Carbon-14 here on earth, and we live in a pretty empty part of space near a relatively inactive star. So I’d imagine the amount of radioactive isotopes of all kinds would generally be higher in more dense stellar regions too.

All that being said, the most common elements will probably be the same. Hydrogen and helium are the most common in the entire universe, and many light elements like Oxygen and Carbon and even Iron which can be produced by fusion will probably be higher in concentration than any heavier elements regardless of where you are in the universe.