Well, there is no catch all answer. It varies by sport.
NHL - EA doesn't actually have any exclusive license. Other companies just don't care to make games on console. Not licensed ones, anyway.
MLB - Sony doesn't have any exclusive license deals here, either. Take-Two used to have exclusive rights to make multiplatform MLB games, but they stopped making them because they found that customers wouldn't buy a trash MLB game just because it was licensed, and eventually the cost of the license outweighed the worth of making the game. Platform owners were always allowed to make their own game even when TT had the exclusive deal, but only Sony bothered. Of course, their game has become pretty lazy and full of microtransactions.
NBA - There is no exclusive license here, either. Take-Two has 2k, but other companies are free to make games. EA has been bringing Live back the last few years. If they wanted, MS and Sony could make basketball games.
MMA - There is no "MMA license". EA has the UFC license. Other companies could easily make MMA games for other brands of MMA, but choose not to. Probably because it would be very niche.
Soccer - FIFA has some license deals but not all of them. PES is licensed to make a lot of leagues. I can't imagine how much it costs to make these games based just on licenses alone, there are tons and tons and tons of different leagues and shit you need licenses for. I believe EA only has the FIFA license exclusively, meaning World Cup. I don't think PES has the US MLS league teams but that's out of choice, not deals. Could be wrong though.
NFL - EA of course has the exclusive license here and it sucks because they've become extremely lazy and football games are dog shit now. This can be blamed on multiple entities though.. EA, the NFL, even Sega for launching NFL2K5 at $20. This lead to Madden dropping to $30 super quick which lead to lower revenue for NFL games which would have lead to lower licensing fees for the NFL, and that's why they made a deal with EA.
Also, there are still a lot of unofficial/arcade type sports games, you just gotta find them. There are plenty for every sport, they are just mostly lower budget digital titles. It's a far cry from generations ago when you could see Blitz and NFL Xtreme on the shelf next to Madden, All-Pro QB, NFL GameDay, etc.
At the same time it's weird with a company like EA. They try so hard with games like NBA and even NHL even though NHL has zero competition. And when they had the NCAAF license, they still made very good games with lots of work put in to make it unique and authentic. Yet with Madden it's just a recycled cash cow for MUT.
Also the "release the game once and then just update it" wouldn't work because these sports companies want money for these licenses. You need a new product every year to pay these fees.