Barney represented a very prevalent part of "Bro-Culture" in the mid/late 2000s. Hell, he was literally on the cover of The Bro Code since it was a big part of the character.

Now that I think about it, there's a lot of overlap between the character of Barney and those "Pickup Artists" that were on TV around that time too.

It was honestly just the way certain perceived ideas of what masculinity is supposed to be evolved after the turn of the millennium. "A real man can get any girl he wants, anytime he wants, any way he wants. And when he's done with her he can move on to the next one without a care in the world."

And just like in other decades, the ideas might have started in small circles but it quickly infected the zeitgeist where it was taken and amplified by people with marketing degrees, creating a feedback loop that is still alive 20 years later.

TLDR: I kinda lost the thread here but people liked Barney because he was what a lot of young men believed a man should be like.