What does the phrase, a person (who) is “a pair of khaki pants inside a Manila envelope” mean?

This is an appeal to the most bland of possibilities. Khaki pants are bland and of an unnoticeable color. In fact Khaki pants were created as part of a British uniform in India. There the sand would blow up and spoil the clean white linens of the marching troops, bad show. Coloring pants with earth tones solves the problem by looking like the dirt. The manila is the same in the stationery world. He is allowing that the candidate may be the least appealing, the least noticeable. With this he would prefer him over the alternative.

I'm pretty sure this isn't a "a common, well-received phrase", rather it is a one-off mixed metaphor. I believe it is combining two contexts where a brownish color implies "plain or neutral, not gaudy". Khaki pants or chinos would be a conservative, not-flashy fashion choice in some circles, and a Manila envelope is a plain brown envelope rather than fancy stationery.

I don't believe there is any connotation of "in upper social hierarchy".

Khaki pants are a rather plain, boring type of pants, and a manila envelope is a rather plain, boring envelope. So, I think "a pair of khaki pants inside a manila envelope" is a humorous way of saying "very plain and very boring."

So, Colbert is saying that if Joe Biden turns out to be a very plain and very boring president, he would like that a lot. This would be by contrast with Donald Trump, whose presidency has been anything but plain and boring.

I think other answers have missed a couple of things

someone who wears “a pair of khaki pants inside a manila envelope”

Firstly notice that no-one wears pants inside an envelope - that would be impossible. The statement says that Biden is pants inside an envelope.

Secondly, the metaphor says that Biden (a) looks boring from the outside and (b) his personality (his inside) is also boring.

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Manila envelope

Khaki pants

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In the light of comments I thought I would take the investigation a step further

Perhaps the phrase is more than just a metaphor. I notice that the colours of the Biden/Harris campaign seem to include khaki. (I'm British so I didn't experience this first-hand)

Here's some campaign merchandise

... and here's a pure guess. Could Biden promotional material have arrived in a manila envelope containing a picture of Joe wearing khaki? I don't know. He certainly does wear khaki pants and even suits sometimes.

Adding to the fine answers already given...

“If Joe Biden is a pair of khaki pants inside a manila envelope, that would be great.”

A plain color, khaki, resides inside a plain envelope.

Colbert may be drawing on several items here.

1. Biden is portrayed as boring, bland, and humble, wrapped in a plain physical appearance.
2. Colbert is happy with boring, bland, humble, and plain after the fireworks of the previous administration. (In his words, that would be great.)

And a couple of things absent.

3. The statement is not so much about Biden being average, even though beige could connote this. (But I admit that there is a humble aspect to beige that may be in play.)
4. I do not think there is an implication about Biden being upper class.

There is a (Southern US) idiom of a yellow dog Democrat. These voters would allegedly "vote for a yellow dog before they would vote for any Republican."(Source: Wikipedia)

Colbert's intro works because he believes American voters have voted for a yellow dog (or a a pair of khaki pants inside a manila envelope) rather than the sitting Republican. But the yellow dog analogy breaks down a bit; rather than choosing out of dogmatic loyalty, the voters are choosing away from the incumbent. This may be why the khaki pants works.

Other answers have done a good job explaining the specific meaning of this phrase, but I think it's worth unpacking how it was constructed.

Metaphors in English generally take "like" or "as," which is more clear, and more common, but not as strong a construction (we call these "similes"). But it's possible to use an unmodified "is" when you want to be more forceful.

The use of two different metaphors piled on top of each other is done to further intensify the meaning for comedic effect. "Khaki pants" and "plain manila envelope" are both relatively common metaphors for bland, unshowy functionality. By combining them together, you reach the concept of something that's almost excessively normal and moderate, which is simultaneously a humorous idea, and a good description of how Biden was positioning himself in opposition to Trump's flashy extremism.

Here's the article referenced in question. The question slightly misquotes the byline

The late-night host is ready for a little less excitement: “If Joe Biden is a pair of khaki pants inside a manila envelope, that would be great.”

That's some key context for the quote. Colbert is ready for a little less excitement. When we look at it in that context, it's clear that he's making a metaphor for Joe Biden. There is nothing exciting about manila envelopes or khaki pants, and by adding the two together, he's engaged in hyperbole. Why is he hoping Joe Biden (then merely a candidate for US President) wins? The third sentence makes that clear

For four long years, Colbert has been making comedy out of the tragedy of Donald Trump. It defined his show, it defined his monologues, it defined his jokes—it defined him. And now, in the excruciating hours of uncertainty, as votes are being counted in Pennsylvania on a Friday afternoon and the nation grinds closer to resolution, the end seems finally, fitfully, at hand.

Colbert has been a public critic of Donald Trump (video contains some strong language) and he's hoping that Biden is someone he won't even have to talk about on his TV show. In other words, he hopes Biden is boring.

This is a figure of speech. It's a variation on a better known quote:

It is a riddle, wrapped in a mystery, inside an enigma.

To convey that [it] is very complicated, it is described as having multiple layers of puzzles and complexities.

Simply put, to express that something is very much [A], you can describe it as "[A1] wrapped in [A2] wrapped in [A3]", where A1/A2/A3 are all types of A.

You can of course use other verbs than "wrapped", as long as you convey that these things are intertwined.

If Joe Biden is a pair of khaki pants inside a manila envelope, that would be great.

Khaki pants are often derided as being a very dull and boring item of clothing. Manilla envelopes are a dull and bland kind of envelope. Their being dull/boring/bland is the common trait that is being pointed out here.

"a pair of khaki pants inside a manila envelope" therefore describes something that is very dull or boring.

So what Stephen is trying to convey is:

The late night host is ready for a little less excitement. “If Joe Biden is very dull, bland and boring, that would be great.”