We need to start with definitions and a common understanding what the words mean.
I like the introduction of the Wikipedia article "Coincidence":
A coincidence is a remarkable concurrence of events or circumstances
that have no apparent causal connection with one another. The
perception of remarkable coincidences may lead to supernatural,
occult, or paranormal claims. Or it may lead to belief in fatalism,
which is a doctrine that events will happen in the exact manner of a
predetermined plan. In general perception of coincidence, for lack of
more sophisticated explanations, can serve as link to folk psychology
So the term "coincidence" describes more a feature of the human psyche to view an event as remarkable - it does not so much say anything about objective reality (let's not talk about whether that exists, now...).
For reality, the more pertinent concept is that of "probability". Some events are more probable, some less so. We know a huge amount of facts about reality that let us put very high probabilities on events. For example, if I pick up a stone and let it go, the chance that it falls to the earth is exceedingly high. It is never 100% - all of the atoms could decide to do some weird quantum jump and end up in outer space. An alien could sneak up and teleport the stone away. A clown might jump out of the bush behind me and grab the stone before it hits the floor. And so on. But in almost 100% of all cases it will simply fall down and lie on the ground.
Is everything just coincidence?
- Some things have a clear causal relationship and behave exactly as we'd expect.
- Some things have a causal relationship but we simply do not see it (yet).
- Some things nave no causal relationship, but we think they do.
- Some things have no causal relationship and we recognize them as such, not even thinking twice about them.
Nothing about this is mystical or even philosophical so far.
Things that we have good reason to expect to happen - because of experience, or because we influenced them to be that way, are not coincidence. Causal chains exists - they are not an illusion. If you put your hand on a hot stove, you get burns, and that will not be a coincidence.
This argument does not care about whether you assume that reality is real or we live in a simulation, or God exists or not. Even if we live in a simulation, there is some code in it that means that if you touch the stove, you hurt. There is no coincidence there.
The same goes for non-human processes as well, from the highest to the lowest levels. Animals, plants, chemical processes, and rocks hurling through the void all follow clear causal chains for most of their behaviour.
There might or might not be "free will", and if you so wish, you can decide to label that as coincidence (i.e., the randomness of quantum processes in your brain leading to random choices). But at this point, we do not know, and you find philosophers who argue pro and contra against that. It is fine for you to simply not decide, or to pick any of the sides.
Doesn't fate exist?
Matter of preference, same as for thinking about what was the root cause of your brain taking a decision with no clear objective reason.
Are all the people meet just met by accident?
Obviously not. If I join a company, it is highly likely that I meet people working in that company. If I am a school child, I will meet my teacher and the other pupils in my class.
You do of course meet random people on the street. If you happen to bump into one of them, fall in love by accident, and spend your next 60 years in deep love => you can decide for yourself if that was an accident or providence. There is no authoritative answer; or rather, there is any imaginable answer out there, claiming to be true at the same time, with none of them having any particular reason to be more true than others.
I think of a cheese sandwich and at the same time I hear someone giving a cheese sandwich recepy on the radio. Is this coincidence? Can this be examined even? Not by looking for a physical connection but by chance analysis?
This depends on your own outlook on reality. If you subscribe to scientific principles, then by Occam's Razor it is coincidence. To prove that it is not, you would do an awesomely complicated experiment and track the occurrence of those two events until you have a reason to believe that there is a causal chain between them.
If you are not so scientifically inclined, you are absolutely free to believe what you will. People do it all the time. Maybe not in the case of cheese sandwiches, but for example when they hear of the death of a person on the radio, while at the same time hearing the dog of the deceased owner barking wildly. The human brain has a very strong tendency to find patterns in the world - this is a great part of our intelligence. Occam's Razor tells me that such a dog-barking event is coincidence. But it's a meme predating the internet that such a thing is not. Whether the one or the other is true is a matter of belief or your own character.