Explore Evolution claims that the common ancestry of life is disputed because:
Biologist Michael Gordon of UCLA argues that the single branching-tree picture of life's history is not accurate, but that life must have had multiple, independent starting points.p. 61
Malcolm Gordon is an expert in the functional morphology of fish, not the origin of life. His argument in "The Concept of Monophyly: A Speculative Essay." (1999, Biology and Philosophy 14:331–348) is that there was likely to be many different environments where life could have arisen (almost certainly true), and that the mobilities of the first organisms would be limited (also likely to be true), that it is probable (note that he never says must as Explore Evolution claims) there would be sufficient time for more than one origin of life event to occur. However, he vastly underestimates the spreading capacity of even slow growing non-motile bacteria. Anyone who has had the misfortune to have a sterile solution contaminated with a few bacteria can attest to how rapidly they grow, and in the prebiotic environment there would be no competition for them. So again, unless the origin of life is astoundingly easy, the first living organism would have had the field to itself. The near universality of the genetic code is consistent with this scenario, but universal common descent does not require a single origin of life, nor have evolutionary biologists ever required only a single origin of life.
There is grandeur in this view of life, with its several powers, having been originally breathed by the Creator into a few forms or into one …Charles Darwin, The Origin of Species, 1859, Final paragraph
As that quotation from Darwin demonstrates, the theory of evolution does not, a priori, demand that life could not have more than one independent origin. However, there are several lines of evidence that compellingly suggest that the origin of life was limited to a very few, most probably one, initial population serving as an "organism." The strongest evidence for this idea is the nearly universal genetic code. This does not argue that the genetic code used by current organisms was found in the first life form; there are several alternative codes that would work. But if there were multiple origins for present-day life, we would expect to see significantly different genetic codes. However, all we see are minor mutational variants of the standard code.
Also, the basic biochemistry of organisms is fairly universal. If life had developed multiple times, we would expect to see at least some organisms with a radically different biochemistry, but we just don't see that.
Furthermore, unless the origin of life was exceptionally easy, it took some time for "living organisms" to develop. Hundreds of thousands to millions of years (even if they were simple self-replicating molecules) were likely required. Once the first living organism had developed, within the space of a few decades its descendants would have monopolized the prebiotic Earth, preventing the development of competitor organisms. It is unlikely, even if the origin of life were astoundingly simple, that a second form of life developed within a few years of the first one to prevent the first's monopoly. Even if a second organism had developed independently a few years after, the first organism with its head start would be much fitter, and would likely out-compete the second.