Well, this gets into a broader scientific problem. It's all anecdotal, of course. There's too much of the Star Wars universe's timeline unexplored to be certain of anything, and the evolutionary or, indeed, prehistoric past of most species - including humans - has not been explored well enough to come to a firm conclusion. If Star Wars' humanity itself had the data necessary to track its own evolution or even home planet, as we nearly do on our own world, we might have a reasonable chance to formulate a likely answer.
Instead, I'm taking a bit of a pseudo-science approach. We know we have X; We know we have Z. Logic suggests there's something Y-shaped between them. Accepting that as a foundation, the core question can probably be summed up with one of two hypotheses.
Many, possibly most, Near-humans and warm-blooded humanoids in general may be based on a common genetic ancestor, though their development is not necessarily due to evolution.
Or...
There is something inherently superior in the bipedal, front-facing sensory organ form that we think of as humanoid which has caused, through natural selection, many species that share roughly equivalent atmospheric conditions to present that specific form.
I personally am a proponent of the former, specifically the idea that most near-humans and, indeed, humanoids in general are based on the genetic information of a human ancestor.
The problem with assuming that it's raw evolution that's generated so many near-human species is that FTL travel at the speed and in the form we know of from the G-Cannon in the Star Wars universe is a relatively young thing by evolutionary standards - only about 25,000 to 10,000 years old at the time of A New Hope. This means that for Near-Human species throughout the galaxy to have evolved from a common ancestor that predates the standard modern Star Wars human is highly improbable, at least based on our current understanding of how a dominant species evolves in a given environment. Over the last 10,000 years on our planet, modern homo sapiens has, in terms of evolution, only managed to triple its average lifespan and we're just now starting to say goodbye to vestigial elements like wisdom teeth.
It's more likely that many near-human species are more likely human off-shoots created by a combination of bio-engineering and selective breeding, either performed on themselves or performed on them by more advanced super-races that have been extant at various times in Star Wars' history. We have evidence of at least one species doing just that - the Arkanians bio-engineered and selectively bred themselves to the point of incredible immunodeficiency.
There's a point in Star Wars history where humans exist and have something akin in power and range to warp drives (if I can reference that other Science Fiction franchise) which they use to travel the galaxy (space travel was not unheard of prior to the discovery of Hyperspace). It's most likely that, as humans attempted to colonize a hostile galaxy, a combination of bio-technology and/or some kind of Force wizardry were employed to accelerate adaptations to bizarre environs. Alternativey, human genetics were used as the basis upon which to accelerate other races by some higher power. Regardless, the human genome certainly seems to be the genetic basis for many species.
We have anecdotal evidence to back this up as well - though perhaps a touch uncomfortable in standard human climes, Twi'leks, Chiss, other near humans, and even Sullustans all respire and operate successfully in oxygen-saturated, nitrogen-rich atmosphere. It's even the standard by which most scientists in the Star Wars universe measure a given atmosphere for its suitability for life. Humans rarely, if ever, concern themselves with atmospheric conditions on a planet when travelling there the first time. Indeed, environmentally sealed suits are only seen in use by military professionals (to counter bio or chemical attacks), to protect severely immuno-compromised individuals (e.g. Darth Vader), or to protect the users from the various dangers of space travel.
Noting the overlap of human foodstuffs being equally valid for humanoid non-humans and even the shared phonology of most near-humans languages (meaning we have the necessary muscles and organs to produce the sounds of twi'lek or other languages), you can come to the conclusion that there appears to be some overlap.
Where that overlap comes from depends on your preference of theory. The alternative to what I've suggested is that the human form, or some variant of it, is inherently superior, leading many other races to evolve into something near enough to it so as to be recognizable by us as humanoid (as you said, convergent evolution).
The reason the theory I've presented is my personal favorite is because I think it makes the story and history of the universe more compelling and interesting. It raises more questions about how, when, who, and why, rather than just be a closed book answer.
