The vast majority of intelligent species in the Star Trek universe are bipedal, humanoid creatures. From the lot I've seen of TOS, TNG, DS9, Voyager, Enterprise, and the movies, it seems that this coincidence goes largely unquestioned.
Metaknowledge grants that it is much easier to have human actors portray human-like beings, and so this was probably done due to budgetary/technological limitations. However, I'm wondering if an in-universe reason for this was ever given or explored?
Yes. In Star Trek: The Next Generation's "The Chase", the Federation, Klingons, Cardassians, and Romulans are all on what amounts to a treasure hunt:
Representatives of all three races are together when all the pieces are put together and it turns out to be a program that takes over a tricorder and projects an image of an ancient alien who tells them that they found the galaxy empty, other than themselves, so they spread seeds of DNA throughout so there would be other races and that the existing races would be descendants of that seeding.
In addition to "The Chase", in the original series episode "Return to Tomorrow", Sargon says that his race had seeded other planets, and that humans and others could be their descendants.
Quoting the Wikipedia article:
Sargon repeatedly refers to the landing party as his children. When Kirk asks why Sargon refers to them as "my children," Sargon says it is probable that human beings actually are descendants of his people. Six thousand centuries ago, they explored and colonized the galaxy just as Earth people do now; he speculates that Adam and Eve were two of their travellers. Dr. Mulhall objects that life on Earth evolved independently, but Spock says the colonist idea would explain certain enigmas in Vulcan pre-history. Either way; Sargon can't confirm either legend.
All this would have happened half a million or so years ago, long after the events referenced in "The Chase".
During the Voyager episode 'Death Wish', the crew is transported back to the big bang by two Q. One Q jokes that if they did stick around their DNA would be spread through-out the universe and perhaps that is how there are so many humanoids. Then he covers his mouth, like it was a slip up that he said it.
The voyager is transported back to the future so it wasn't their DNA that was spread. But the idea seemed interesting that pieces of DNA could be spread more or less ubiquitously over the entire universe.
In TOS, there was a race called the Preservers in the episode The Paradise Syndrome. They transported groups of endangered indigenous people across the galaxy and terraformed the planets as well (a possible explanation for all those earth-like planets).
TNG The Chase, was written by Ronald D Moore. In the Star Trek TNG Companion he mentioned that he intentionally did not specify that these aliens were The Preservers and left it up to the viewer to draw the connection.
The Preservers are mentioned quite often in the trade paperbacks, and it's generally assumed that seeding is how the galaxy was populated.
The extraterrestrial seeding theories are all a bit suspect to me from a scientific point of view (besides the Adam and Even nonsense from Return to Tomorrow).
We already know that modern humans descended from primates which have been on earth for 65 million years, and their evolutionary roots can be traced back to the common ancestor of all mammals, the Synapsids, which lived 250 to 300 million years ago. So any exogenic explanation for the origin of humanity has to be set much earlier than half a million years ago. And the "seeds" themselves would have had to been single-celled organisms to explain what we know from modern genetic analysis of life on Earth.
A more plausible explanation is one that attempts to explain why so many species on different alien worlds have near-identical analogs on Earth (hawks, dogs, lizards, beatles, worms, etc.), and that would be convergent evolution.
It's possible that due to similar environments existing on different worlds, evolution has independently found the same "solutions" in terms of general body layout and organ function. Physics (for the most part) follows the same rules no matter what planet you're on, so that in itself is going to eliminate certain body designs while gravitating towards others.
That would explain why most complex life forms are bilateral-symmetric and only a few are radial-symmetric. It also explains why most animals have an even-number of limbs and heads are placed at the front or top of the body.
So the reason why humanoid species are so common in Star Trek is probably because the humanoid body shape is the natural upright evolution from a quadriped bodyshape, and it's one that allows one to use their hands for manipulating the environment (e.g. creating and using tools). And we don't see any sapient aliens with 2-mouths or eyes on their back because it's just inefficient to have these designs.
My understanding - although I cannot quote particular evidence - is that the seeding argument is the reason for this. I think that the understanding is that some species was responsible at core for seeding the entire galaxy. This includes earth, and explains why many creatures have the core bipedal approach to life.
Of course, many of them have evolved in a range of ways to make them more of less humanoid. Most have forgotten that there was a seed race, some have stories that explain it. I think the question of who the initial seed race was is deliberately avoided - not that anyone actually knows for certain. It may be extinct except in its many seed forms.
This is explained in The Paradise Syndrome, the 25th episode of Season 1 of The Original Series:
MCCOY: Were you able to make sense out of the symbols?
SPOCK: Yes. The obelisk is a marker, just as I thought. It was left by a super-race known as the Preservers. They passed through the galaxy rescuing primitive cultures which were in danger of extinction and seeding them, so to speak, where they could live and grow.
MCCOY: I've always wondered why there were so many humanoids scattered through the galaxy.
SPOCK: So have I. Apparently the Preservers account for a number of them.
