Extremely low possibility, but if life synthesizes from scratch after life has existed on a planet for a few million years, would the two separate lineages be able to co-exist?
1/2: Shadow Biospheres
Encyclopedia Britannica's article on abiogenesis claims that some scientists support the theory of multiple geneses on Earth:
Some scientists have proposed that abiogenesis occurred more than once... phosphate-based life... gained an evolutionary advantage over all non-phosphate-based life (“nonstandard” life) and thereby became the most widely distributed type of life on Earth. This notion led scientists to infer the existence of a shadow biosphere, a life-supporting system consisting of microorganisms of unique or unusual biochemical structure that may have once existed, or possibly still exists, on Earth.
Such a shadow biosphere, though discredited due to a general lack of evidence on Earth, is theoretically possible. For a planet on which two chemical types of life are generally equally fit to survive, perhaps neither biosphere would be a "shadow" - both could exist independently as long as the environment remained suitable.
2/2: Differentiated Environments: Life on Titan
It might be easier to get multiple biospheres on differentiated worlds with vastly different, isolated environments. Let's take a look at Saturn's moon Titan. Although this world is famous for its hydrocarbon oceans, astrobiologists are equally fascinated by its subsurface liquid water ocean.
Many scientists believe life on Earth formed in hydrothermal environments in Earth's oceans - providing a mechanism to populate Titan's subsurface (if that high-pressure ice shell doesn't get in the way). Meanwhile, others have tested the theory that life formed closer to the surface in the presence of lightning - providing an unlikely, yet plausible means of populating Titan's surface oceans (if hydrocarbon seas can even support life).
Imagine a world with methane seas populated by methane fish - swimming miles above an icy, dark abyss teeming with vastly alien creatures. It's unlikely, it's speculative, but it's cool as hell.
Instead of going straight to different materials, let's first look at different ways that life can manifest with the materials it already has. One way to do this is through Chirality.
Chirality in chemistry is basically a description of the asymmetry of molecules, and organic molecules all seem to share a specific chirality which we call 'left handed'. We know that mirror molecules can exist, but they don't seem to naturally occur.
It is believed that they used to however and there are several theories about why our chirality became dominant which you can read about if you're interested. The key point about chirality is that even though it doesn't involve a different materials, it does demonstrate that different formats can be incompatible and that one eventually rises above the other if they share a given ecosystem.
Of course, sharing a given ecosystem is even harder if life formed from different materials is trying to do it. For instance, life that oxidises Fluorine instead of Oxygen is going to find any oxygen rich environment toxic, and vice versa. We as oxygen breathing life forms find many flourine based compounds highly toxic and dangerous to our health. Life based on (for instance) a metal based element with high valency like Chromium is going to struggle to compete for the other elements like Hydrogen, Oxygen or even Iron in a shared environment because carbon is more plentiful and is likely to bind up more of these elements than the available chromium, meaning that it will be harder for chromium based lifeforms in such an environment to find molecular energy they can metabolise.
In short, the odds against this are extreme, largely because of just how organic chemistry works. When you get right down to it, different formats of life are going to be in open competition with each other for available resources in such an environment, and ultimately one of them is going to win it at some point in the evolutionary cycle. Once that happens, the loser is likely to die out completely, leaving a single dominant life format.
The amount of balance required in a complex ecosystem to stop that dominance happening at some point would just be too sensitive to be maintained in my opinion.