Where we think* it’s likely to form. We’ve yet to find verifiable genesis elsewhere, and so we can’t well determine whether the nature of our genesis is common or rare.

However, there really aren't many candidates for systems of complex chemistry that don't primarily revolve around water and carbon. Water's role as a 'universal solvent' that's not so aggressive that it tears other molecules apart is extremely important for life to be able to form, and it's very difficult to replicate that effect with any other chemical.

And not even worrying about liquid water for a moment, the temperature range itself could be crucial to life forming and operating. At very low temperatures, chemicals tend to react slower, to the point of becoming overly stable and inert, so it's difficult to get any complex reactions going that could lead to life. At very high temperatures, chemicals react quickly and become unstable, so it's difficult to get any complex molecules to stay together more than a few seconds. Even without worrying about the chemistry of water, there's a relatively narrow range of temperatures where the extremely complex chemistry of life is possible.

Maybe it's possible for life to form and exist outside the normal 'goldilocks zone' conditions we think of as favorable for it ... but the deeper you look into it, the less and less likely that seems.

And how many times has life started on Earth alone?

Only once that mattered. It has surely started countless times ... but every time after that first one was extremely short-lived, because whatever primitive proto-life managed to develop in subsequent times was quickly out-competed and/or eaten by the life that had already developed, evolved, and become much more advanced.

We can be sure none of these other starts ever led to lasting lineages by looking at genetics and the chirality of certain organic molecules. All living things on earth share at least some genes in common, and their DNA (or RNA) work the exact same way, with the exact same chemical structures. For organic molecules that can either have left-handed or right-handed chirality, all living things on earth produce and interact with the same left or right-handed versions, never the opposite chiral pairs ... even though it was only up to chance which one of them developed first. (DNA itself is one of those chiral molecules -- in every organism on earth, the double helix of DNA twists in the same direction, never the opposite direction.) There's no fundamental reason for life to prefer one direction over the other; if life had started multiple times and led to multiple lineages, we'd expect to see each lineage having different chirality preferences.

Similarly, our definition of life may need changing upon alien discoveries.

"The property or quality that distinguishes living organisms from dead organisms and inanimate matter, manifested in functions such as metabolism, growth, reproduction, and response to stimuli or adaptation to the environment originating from within the organism."

Nah. If it doesn't at least mostly match that definition, it's not alive.

Though if we ever manage to make self-replicating robots, we very well might have to revisit that definition ... or accept that the robots are alive.