It is for financial reasons.
In the past, many series had longer seasons: 24, 26 or 28 episodes for a season was common in the 90s (Meitantei Conan's first season was 28 episodes). Series that were pretty guaranteed to be popular enough to keep up selling a lot of toys to kids had longer seasons (Sailor Moon's first season had 46 episodes). Gag series from 4-koma comics are more likely to have a long series length (Nintama Rantarou's first season had 47 episodes and Chibi Maruko-chan's first season alone was a whopping 142 episodes). In the past, more people were loyal to their favorite series and collected the merchandise carefully, whereas Japanese children now are quick to switch their fads (from Pokemon to Youkai Watch, for example), so investing a lot into a series that'll get discarded next year is not financially-sound.
12- or 13-episode series started to appear in the late 90s as a way to animate and broadcast something (adapting a manga or video game, or an original series) that did not have a big enough fan base to sustain financing a full-length series (of course, this was in the era when everything was still drawn by hand). Previously, those sort of anime only received a one-shot OAV or a brief OAV series (for example, Koko ha Greenwood or Tokimeki Memorial).
Even though anime is mostly animated outside of Japan and by computer, lowering production costs, the industry is able to further expand the genres and number of titles produced by switching to a lot of shorter series over a smaller number of long-running ones. To have a long run, a series needs to keep viewers coming back (the sort of plot that can go on and on, like Inu Yasha) and sell merchandise well. Many genres of manga do not lend themselves to those epic series, so studios are creating shorter series that:
- Can get something animated which in previous generations of anime production would never have had a chance to get animated at all because it's more niche.
- They can make a modest amount of money off of it if it does well (and give it another season), or they can chalk it up to not too drastic of a loss if it doesn't do well and quickly move on to the next one.
- Provide a nice, tight story arc that has closure at the end (though when they are not sure up to nearly the last minute whether they will get the green-light for another season, the ending can still fizzle).
Although the main reason is financial, a series where the very limited total number of episodes available to work within is known to the staff from the start can allow the writers to craft the storyline to suit the length with little to no filler, which offers a different sort of project than Rurouni Kenshin or HUNTER x HUNTER where you don't entirely know where it's going and how many episodes it will need to get there.
I'm not sure what you mean by "the seasons themselves don't much align to actual seasons, with 'Spring 2015' ending in late June." Japanese culture is very specific on when seasons take place, what is associated with that season, what to do during each season, what day of the year to switch from spring/summer school uniform to fall/winter school uniform and vice versa (called koromogae: see 2.07 here), what to eat in which season, etc. The seasons coincide with the Japanese academic year, so that the sakura (cherry blossom) trees are blooming in most of the country during graduation ceremony in late March and during the first day of school (on April 1st). This is a primary reason why, although Tokyo University has switched to the Western academic year, most schools in Japan strongly do not want to. Anime is usually keen to let the viewer clearly know what season it is in the story, and try to coincide this with the air date of the episode (New Year's Day, St. Valentine's Day, sakura blooming, the sound of cicadas in summer, mosquito repellent, summer festivals, ishi yaki imo (stone-baked sweet potatoes) in autumn, fall leaves, snow, etc.).