I believe it is theorized that life could use silicon instead of carbon, but I think for the most part we just look for what we know works. They are just saying that earth life could have lived on Mars at some point in the past, that the conditions would have been favorable for it. Other kinds of life could exist in other environments, but we don't have any litmus for those observations yet.
Life could perhaps also just use a computer, and exists entirely as software. The computer could be implemented with something other than atoms, perhaps. Maybe some particles shooting among the neutrons of a white dwarf star or something. (My point here is that life doesn't have to be chemical, based on the elements and compounds.)
However, for chemical life, there are indeed only a few ways. We understand the elements very well and the kinds of combinations they express: what kinds of chemistry are possible for all the elements.
Only a few elements exhibit the expressive variety of molecular forms that plausibly support the complex structures that are arguably needed for the chemical representation of life.
Presumably such a computer would have to have been created by some extremely complex self-organizing system. I think we would call such a system "life" long before it was complex enough to turn a star into a computer (ignoring the thermodynamic problems with such a proposal).
The problem is that silicon dioxide is a solid unlike carbon dioxide.
I'm not a chemist, but by analogy with anaerobic bacteria we have on Earth, some of which produce methane gas, one could imagine something that expels silane gas.
Because all organisms need to expel entropy to survive, and expelling solid entropy is a fairly non-trivial engineering problem vs. the life that we do know expelling it via liquids and gasses. Heck, it's not even all that clear how to do that just as a machine... nothing is coming to mind that does that today. (Examples welcomed. Also note that for today's discussion execrement is solids suspended in a liquid, not a "solid". Silicon is a solid, you know, rock-solid, literally.)
The idea that we have "no idea" what life could look like out there in the universe is grossly oversold. We actually can put a lot of bounds on them, using rigorous mathematics, and discuss them despite not necessarily knowing all the chemical details.
It's difficult to see how this could be made to work for a life form, because this is a very macroscopic process. On a microscopic level, well, we're all just ugly bags of mostly-water so we do it by suspending things in our mostly-water. As I understand it there are few to no good candidates for a silicon-friendly liquid base when you work through it.
(Interestingly, I have seen some suggestions that there could be other liquids that carbon-based life could be based on, such as ammonia. That said, it may not be coincidence that water is still almost certainly the best (accounting for the possibility that it's just the observer effect), and that's what all life forms reading this for the forseeably future are based on.)
The idea that we have "no idea" what life could look like out there in the universe is grossly oversold. We actually can put a lot of bounds on them, using rigorous mathematics, and discuss them despite not necessarily knowing all the chemical details.
