Let the brother of low degree rejoice in that he is exalted: But the
rich, in that he is made low: because as the flower of the grass he
shall pass away. (James 1:9-10, KJV)
In understanding James' usage of "brother" in this specific verse, it helps to see how he uses it elsewhere. Context is always a valuable reference point when studying the Bible.
James uses "brother" and "brethren" multiple times. His opening salutation addresses the "brethren."
James, a servant of God and of the Lord Jesus Christ, to the twelve
tribes which are scattered abroad, greeting. My brethren, count it
all joy when ye fall into divers temptations; (James 1:1-2, KJV)
Clearly, the "twelve tribes" are a reference to the people of God, and not to all mankind generally. Therefore, James' usage of "brother" or "brethren" follows in this context. In Chapter 2 we see this again.
My brethren, have not the faith of our Lord Jesus Christ, the Lord of
glory, with respect of persons. For if there come unto your assembly a
man with a gold ring, in goodly apparel, and there come in also a poor
man in vile raiment; (James 2:1-2, KJV)
Here we see that the "brethren" are addressed as having faith in Jesus Christ and being among those who assemble together. Here also we see a contrast made between rich and poor. Throughout the book of James there is a considerable interest in rich versus poor, continuing through to the last chapter where the rich men who have cheated (KJV: kept back by fraud) from the poor are soundly rebuked. In this chapter, however, no reference is made of these rich men being brethren. This is significant, and instructive.
In James 1:10, James clearly references a fellow believer by the term "brother." How could James properly tell an unbeliever to be humble? Which unbeliever would be reading James' letter, much less following its Christian counsel?
The point that seems to be misunderstood by some and which leads them to an incorrect conclusion as to the identity of the "brother" in the verse is the next part.
...because as the flower of the grass he shall pass away. (vs. 10b)
This follows another repeated theme in James' writings: the ephemeral, mortal nature of man. Furthermore, it is a quote from Isaiah.
The voice said, Cry. And he said, What shall I cry? All flesh is
grass, and all the goodliness thereof is as the flower of the field:
The grass withereth, the flower fadeth: because the spirit of the LORD
bloweth upon it: surely the people is grass. The grass withereth, the
flower fadeth: but the word of our God shall stand for ever. (Isaiah
Peter also picked up this theme from Isaiah:
For all flesh is as grass, and all the glory of man as the flower of
grass. The grass withereth, and the flower thereof falleth away: (1
Peter 1:24, KJV)
This is not meaning that every human will be lost forever. This is not a reference to salvation so much as to the ephemeral nature of man. As David said:
As for man, his days are as grass: as a flower of the field, so he
flourisheth. For the wind passeth over it, and it is gone; and the
place thereof shall know it no more. (Psalm 103:15-16, KJV)
"Grass" is a Biblical symbol for humans because both are short-lived.
In conclusion, yes, the rich man referenced in James 1:10 is a brother, and he is counseled to be humble knowing that his mortal life here on earth will not last forever.