Another possibility is that the rivalry is mutual, but refusing to acknowledge it is Bob's way of messing with Alex.
Alex eventually witness Bob's prowess when he get serious, realized that he Can't Catch Up with Bob and drops their rivalry.
Now that Alex has proven himself he no longer considers Bob to be a rival.
Alternatively, Bob pretends that the rivalry is a mutual rivalry to make Alex feel better.
It look like Bob finally consider Alex as his rival, but soon turn out that he misremember him for someone else.
Alex eventually witness Bob's prowess when he get serious, realized that he Can't Catch Up with Bob...for now. So he goes on training even harder, so he can challenge Bob as Worthy Opponent.
An increasingly frustrated Alex repeatedly tries to jog Bob's memory by mentioning all the times he has fought Bob by referring to them in great detail. Bob still says that his name doesn't ring a bell.
Bob doesn't remember ever fighting Alex, but he mentions that he can vaguely remember fighting Alex's accomplices, Charles and Dylan, under the circumstances Alex describes. Bob also mentions that he thinks that there might have been a fourth guy involved in the fight, but that could just have been his mind playing tricks on him.
Zig Zagged: Bob can't decide whether or not to take Alex's claims of rivalry seriously
Averted: Alex and Bob are mutual rivals.
Lampshaded: "It's like Bob doesn't even know I'm his rival or something!"
Alex: "Bob, now I shall put the end to our rivalry!" Bob: "...Who the hell are you?"
Alex's constant efforts to make Bob acknowledge him as a rival are the result of an unhealthy obsession with Bob that is slowly destroying his life.
Alternatively: Bob doesn't see Alex as a threat, and thus underestimates and ignores him. Alex, who has been obsessed with defeating Bob, has spent some time learning Bob's fighting style and his weaknesses, and kicks his ass as a result.
Played For Laughs: Alex engages in a Zany Scheme in an effort to force Bob to acknowledge him.
Played For Drama: Alex becomes more and more vicious, angry, and ruthless as Bob continues to ignore him, until he reaches the point where, mad with envy and out for blood, he's a very real threat, whether Bob knows it or not.