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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:
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.
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.
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.