I think the reasons he's popular in the fandom include the following:
1. He's got a sense of humor. In Bioware's fanbase, this tends to override a lot of traits. Notice that one commonality with all the villains in Bioware games? No sense of humor. If a character has a noticeable sense of humor, we're supposed to like them. But if you're not fond of overt snarkiness and cracking jokes, Varric will probably not appeal.
2. He's not judgmental. Man, woman, dwarf, human, elf, blond, redhead, mage, lyrium-infused, bloodmage, templar, most of it doesn't matter to Varric, at least not in DAII. He takes people as they are, doesn't demand things of them, seldom asks favors, and usually strives for mutually beneficial arrangements. This tends to make a guy a likable person to most people because he doesn't offend others, at least not without good reason.
3. He's not self-absorbed, instead focusing on his friends, including the PC. As much as Varric talks, we seldom hear him say things of note about himself. He's mostly about other people. He helps Daisy adjust to the alienage and keeps the street thugs away from her. He keeps the Carta off Blondie. He tells Hawke's story, not his own. He helps other people, even his traitorous brother when it comes right down to it. When did Hawke ever ask Varric to do something and help with something, and Varric said "Nope, I won't do it."? I can't recall it happening.
4. He's loyal. No matter what happens. Side with mages or templars, be a total a**hat or the epitome of selflessness and virtue, Varric stands at your side through everything. He may even dislike Hawke in many ways, but respects him/her to a fault. This kind of loyalty is difficult not to find endearing for most people.
Those are all traits that most people like to have in friends. And having them all in one particular friend is the proverbial jackpot. Friends like the above are hard to find, and tend to be the cream of the crop. Hence, Varric is popular.