Take modern art, for example. People draw a circle and call it art, and apparently it automatically has a copyright after it's made. Then I come along and draw another circle and call that art, and it looks very similar. Would that be against the law because someone has already made it before, or can I use it if I have proof that I didn't copy their art? And keep in mind I don't know much about this because the only law experience I have is from Wikipedia.
This sounds like a case where there can be differences as to if something is a design that can be copyrighted or if it is a matter of a trademark held by the artist who drew the circle.
If someone began a copyright infringement action against you, claiming copyright of the shape of a circle, your defense could be the Threshold of originality (Wikipedia).
The threshold of originality is a concept in copyright law (Wikipedia) that is used to assess whether a particular work can be copyrighted. It is used to distinguish works that are sufficiently original to warrant copyright protection from those that are not. In this context, "originality" refers to "coming from someone as the originator/author" (insofar as it somehow reflects the author's personality), rather than "never having occurred or existed before" (which would amount to the protection of something new, as in patent protection).
Obviously, circles, squares, triangles, pyramids, etc., even straight lines, have all existed for a long time.
Simple shapes can, of course, be included as part of a more complex design, and that design as a whole is copyrighted as soon as you complete the design. You don't need to claim copyright; you already have it.
If you did a design that was fairly simple and were accused of copyright infringement, it might come down to a court case to determine the threshold of the threshold of originality.
Now, if you did a painting of a circle that looked like another painting of a circle, then you could be accused of trademark infringement, because the artist you allegedly copied could say that they have such a distinctive style that is is their trademark.
A trademark, trade mark, or trade-mark is a recognizable sign, design, or expression which identifies products or services of a particular source from those of others...Trademarks (Wikipedia)
Trademarks are also designs, of course, but are protected in a different method different than copyrights.
So many decisions about copyright infringement come down to the judgement and opinions of a jury, because of the fungibility of what exactly constitutes originality and what constitutes a copy, part of a copy or a derivative.
Trademark infringement hinges on how recognizable that design as a mark is and if there can be confusions between the two.
For a much more in depth outline of copyright, see I have a question about copyright. What should I read before I ask it? - Law Meta Stack Exchange.
A copyright only protects an original work of creative authorship. A circle, by itself, has been public domain for centuries and cannot be protected as "original" or as "creative" in any sense. Nobody owns copyright on works in the public domain. They would have to add something "creative". They would then have to prove your "identical" work was a result of your having had access and making a copy of theirs. Simply having two identical works does not prove "copying".
Consider: Two photographers stand next to each other, overlooking the Grand Canyon, holding identical cameras with identical settings. A large bird swoops by and they point their cameras in the same direction, with the same focus, and snap pictures simultaneously. Millions of people may have similarly photographed that very spot, and the circumstances of lighting, framing, speed, weather, and capture of the bird swooping might make both of these photographs "creative" and "original", if not "identifiable".
By law, they each own the copyright of their respective photographs, not to mention ownership of the original. Either may distribute copies free of interference by the other. Either or both may register and enforce their copyright against others. The mere fact that they are effectively identical (including time-stamp, etc) is not proof that one is an unauthorized copy of the other.
To be protected by copyright law, a work most be original and exhibit a modicum of creativity. Only the expression of some underlying idea can be protected. So no copyright exists for "drawing a circle", but a specific drawing of a circle can be so protected. If you exploit the same underlying idea ("draw a circle and call it art"), your specific expression of that unprotected idea is also protected. However, if you meticulously copy an existing expression of that idea, that is infringement.
As for the issue of proving that you copied, if you were to draw a circle that was substantially similar to one of the extant "just a circle" paintings" such as Damien Hirst "The Black Sun" or Kazimir Malevich "Black Circle", there is enough else that isn't "just a circle" that actually copying could reasonably be suspected. Whereas if your work was just a perfect circle with no fancy fill (just solid black), that would not be sufficiently similar, in copying the creative elements of the existing works. Presumably, a pure machine-drawn circle with a solid fill would not pass the creativity test, and would not be protected.