Largely because of the personalities, and during that time, egos in involved with 'The Kliq'. HBK, Hall/Nash, Triple H, they were not only seen as a unit by smart fans, but they were guys that were legitimately pulling strings behind the scenes.
Bret, Own, and Bulldog were of course seen as a collective in a lot of ways, but that was based largely off the fact that they were family. It wasn't until Bret formed the his heel stable Hart Foundation with Bulldog, Pillman, Owen...that it really became something WWE embraced.
HBK was at the height of his first WWE run, and at the height of his personal demons. Triple H was coming in to his own as singles wrestler, and by virtue of his friendship with Michaels, was sowing the seeds to become a force behind the scenes. Hall and Nash, were Hall and Nash.
Now, I'm not sure what you are trying to tie in to your first question, when you ask about the match quality between HBK and British Bulldog. If you're asking if their match quality had anything to do with WWE recognizing BB, Owen and Bret as a 'kliq' the answer is no. HBK, BB, Owen, Bret, they all had great matches with each other, but none of those matches really factored in to The Kliq or Bret's allies in terms of their perception or popularity as units.