I really hope the "competition" is a competition between different guild branches for some sort of bragging rights, and that they just picked the MC to be their representative, and they have to hurry to get him to replace their previous one. Since the old man talked about the prestige of the guild, it might be that they have come in last place the last [insert number] of tournaments, and he is looking for somebody to turn things around.
@feha I would say this in an issue of basically "freelance adventurer" vs. "affiliated adventurer". After all, she asks him if he can give proof of achievements, to prove he is worthy to be an adventurer. Proof of monsters killed would seem to be the easiest thing to provide. If he had shown her proof of the cockatrice he had killed, he probably would have been immediately admitted, but he didn't, because he didn't consider that an "achievement" because of the levels of his accomplishments in his previous lives (and because of being incredibly dense, as shown with the skills just afterward). The fact that he is asked to prove it with "achievements" not "previous registration" shows that he doesn't have to be previously associated with a guild branch to register as an adventurer at this branch.
I think this is basically how most settings with "adventurer guilds" work, and that being "affiliated" has a lot of benefits. It allows them to get jobs that are given to adventurer guilds, which probably have better rewards. It also allows guilds to become convenient places for anybody to post jobs, and have a better chance of somebody taking it (with the guilds having a certain standard, perhaps, for what a job should cost). It allows guilds to have a better idea of the strengths and abilities of various adventurers, both for the assigning of tasks, to try to make sure they are completed (which boosts the guild's reputation, as well as the adventurer's), and for knowing what might be needed to defeat them if they get out of line and start causing trouble. This also makes them valuable as a way to keep adventurers in line, and not going berserk or terrorize/extorting townspeople.
He also could have shown his skills to become accepted as an adventurer (as they would take those as proof of being qualified), but he decided that, when he isn't sure if they count or not, the solution is to not show them and find out, but just to assume they don't count, even though the few experiences he has had would have shown they should. Frankly, the big issue is how dumb he seems to be in evaluating the state of this world, even though he had previously risen to the top in two different fields (and magicians are normally expected to have a decent amount of intelligence). I guess authors decide this is an easy way to build tension (or maybe comedy), but it often isn't executed well, and tends to take away from the flow of events, and cause frustration for readers.