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I found this story on the web several years ago, lost track of it, and can't find it now. It's a kind of different isekai story where, instead of the standard formula of our protagonist being summoned to a magical world and greeted by someone who tells him he's the Chosen One who's needed to face some threat, this guy just ended up in a magical world one day with no idea how he got there or why.

He doesn't have much in the way of skills that would be useful in a medieval-tech-level society, so he has to rely on his wits, and on a unique power he has to break or suppress magic. Unfortunately he's in a civilized, peaceful kingdom where most of the things that that would be useful for are illicit dealings, so he ends up becoming a thief out of necessity.

He figures if it could happen to him it could happen to someone else too, so he tries to get the word out by having bards compose a catchy song about him that he's hidden a bunch of pop culture references in, so anyone from Earth who hears it will know to look for him. I think one of them was about Mario and Luigi. Really early on, he finds out it worked, that there's someone else from Earth who was there before him, who doesn't break magic; she's actually a big-shot wizard, but he's never heard of her because she lives way off in another kingdom. He promises her servant who was sent to find him that he'll go visit her when he can get around to it, because for some reason she can't come in person.

He gets caught stealing something from someone powerful, who instead of punishing him twists his arm into taking on a big quest to steal something from a dragon, which he totally thinks is a bad idea but he doesn't have much of a choice. He gets sent along with a companion to keep him in line, a warrior-woman who he totally has the hots for but of course he doesn't want to admit it to her. I think the quest was going to take him close enough to where the wizard lived that he could visit her along the way. And that's about as far as I remember reading.

That sounds like you got about a third of the way through The Lay of Paul Twister, the first book in the The Tales of Paul Twister web serial. Calling it "isekai" is a bit misleading, as this is not in any way a Japanese story--the MC is from Seattle, all the pop culture references are American, (or stuff Americans would recognize, such as Shakespeare,) etc--but otherwise this is pretty close.

It's a kind of different isekai story where, instead of the standard formula of our protagonist being summoned to a magical world and greeted by someone who tells him he's the Chosen One who's needed to face some threat, this guy just ended up in a magical world one day with no idea how he got there or why.

Check:

And then one day, I was just driving to the store, about two miles away, and suddenly I wasn’t anymore; I was in the middle of a forest. And I wasn’t even going 88 miles per hour! (Luckily, I was doing about 25 or so. Much faster and I’d have smacked into a tree before I could bring it to a stop!) I had no clue what had happened. Neither the GPS nor the radio could pick up anything, and my phone had no reception.

One thing I could tell for sure, once I got out of the forest, was that I was nowhere near Seattle anymore, past or present. You know those really steep hills sloping down to the waterfront that San Francisco is famous for? Seattle has them too. But wherever I was, it was clearly inland, not coastal. I burned a fair amount of my already-limited supply of gasoline heading towards the setting sun, trying to find the waterfront as a way to orient myself. I’m just glad I noticed I was approaching a peasant village before any of the peasants noticed me! I had to ditch the car somewhere no one would find it, and make my way into town on foot.

(I should note that Paul does eventually learn how he got there and why, but as the series goes on it only gets more and more ambiguous as to whether he was "chosen" in some sense or if him ending up in the other world was just something that randomly happened to him.)

The magic-breaking power is a mysterious ability known as "the Twist" that only Paul has. He names it (and himself after it) when the first person to try to use magic on him complains about him "twist[ing] my magic!" when it fails.

he tries to get the word out by having bards compose a catchy song about him that he's hidden a bunch of pop culture references in, so anyone from Earth who hears it will know to look for him. I think one of them was about Mario and Luigi.

The titular "Lay of Paul Twister". The song doesn't mention Mario and Luigi, but does mention how

he sneaks into a dragon’s keep to rescue a princess, but finds that the dragon tricked him, and she was being held in another castle entirely.

The part where

He gets caught stealing something from someone powerful, who instead of punishing him twists his arm into taking on a big quest to steal something from a dragon

is really where the main plot gets kicked off, except the dragon in question died of disease long ago and the thing that needs to be "retrieved" is the dragon's skull, because it will be useful to somebody for magical reasons.

He gets sent along with a companion to keep him in line, a warrior-woman who he totally has the hots for but of course he doesn't want to admit it to her.

That would be Aylwyn, an angelic paladin and Paul's main love interest. Of course, with her being waaaaaay on the Lawful Good end of things and him being a thief... yeah, he figures nothing good would come of telling her how he feels.

I think the quest was going to take him close enough to where the wizard lived that he could visit her along the way.

That's actually one of the first things he thinks of when he realizes where he's being sent. Of course, in Paul Twister stories, things never quite work out the way he plans... but I won't spoil it for you.

And that's about as far as I remember reading.

Wow, you lost track of the story just as it was really getting good. The author got through three novel-length stories and partway into a fourth before really slowing down due to RL taking over his free time. The most recent chapter was posted July 2017. Hopefully there will be more soon!