Why is there only one kind of biochemistry on Earth?

One piece of evidence worth knowing, because it is the strongest thing we have: the shared arbitrary choices.

All known life uses the same handedness of amino acids and the same handedness of sugars, from two chemically equivalent options. All known life uses substantially the same code mapping genetic triplets to amino acids, out of an astronomical number of possible mappings.

Those are arbitrary. There is no strong chemical reason to prefer one handedness, and the code's specific assignments are not forced. Universal sharing of arbitrary choices is exactly the signature of common descent rather than of convergence — the same argument that tells you two documents were copied from one source rather than written independently.

So whatever happened at the origin, everything alive now is one family. That does not rule out other origins having occurred and lost.

There is an active line of research that follows directly from your question: looking for a shadow biosphere — organisms with different biochemistry, surviving in places our incumbent life does not reach or in forms our detection methods would miss.

The methodological problem is severe and worth appreciating. Almost every technique for finding life looks for the biochemistry we know: our genetic material, our proteins, our metabolic signatures. Something built differently could be sitting in a sample and be invisible, because we are running the wrong test.

Nothing has been found. Given the detection problem, absence of evidence here is weaker evidence of absence than usual, which is why the idea remains taken seriously rather than dismissed.

The honest answer is that we do not know, and there are three candidate explanations that are not mutually exclusive.

It only happened once. The origin of life may be so improbable that it occurred a single time and everything descends from that event. Under this view there is nothing to compete with because nothing else ever started.

It happened several times and one lineage won. An established biosphere is a hostile place for a fragile new one — the incumbent eats the resources, and quite literally eats the newcomer. A second origin today would be consumed almost immediately by organisms that have had billions of years to get good at consuming things. This is sometimes put as: any new life would be food.

The chemistry is genuinely constrained. Work on alternative replicators suggests that some candidates are markedly worse at the job — less stable, less accurate, harder to assemble. If the chemistry has a strong optimum, convergence rather than accident would explain the uniformity.

My own read is that the second is doing most of the work, but that is a judgement rather than a result.

Worth adding what the question would look like if answered elsewhere: this is one of the main scientific motivations for looking for life on other bodies.

A second, independent origin found anywhere — even something very simple — would immediately settle whether life's chemistry is forced or accidental, because we could compare. One sample is a mystery; two is a science.

Which is why the interesting question about a subsurface ocean is not "is anything alive there" so much as "if so, is it related to us".