The older answer isn't wrong by any means, but it only covers the fact that the music for R/S/E is chiptune (I'd highly recommend you read it if you're curious about the specifics regarding that.) I would, however, like to give a more in-depth explanation that's a bit more... musically informed and nuanced. I'll address the music of the Pokemon games as a whole, but I will specifically try to gear things more towards the music of R/S/E.
I'd hesitate to define any music as "chiptune." Wikipedia actually defines chiptune as a style, rather than a genre, while it specifies that genres are... genres (take, for example, the page on rock music states "genre" rather than "style"). I concur with this statement, as chiptune is more of a set of sounds, instruments, if you will, that can be used to play pretty much anything. Similarly, you don't ask a trumpet player what music they play and get an answer like "I play trumpet music!" Defining something as "chiptune" tells you the method, but it doesn't actually tell you what the music actually is. This is common sense if you think about it - you can play classical music on an electric guitar with crazy distortion and it's still classical music (despite the instrument and sound being a bit unorthodox for the genre). Likewise, you can have thrash metal classic BYOB and Pachelbel's timeless Canon in D both in chiptune and neither are the same genre (though this raises the question if these songs remain as parts of their initial genre. This is above my paygrade, so I will be ignoring it). Let's apply this to the situation at hand: Compare the original Victory Road theme and the remake. These are really similar. Pretty much the same instrumentation, just with an updated soundfont (sounds to me like it was recorded by an actual orchestra, but in the 12 years between the games, there have been a couple advancements in technology, so it may all be digital). You can define the original as chiptune, but not the remake. If you're after the specific sound of the music, look for more chiptune music, but if what you like about the music is the general sound, how instruments interact with each other, you'd probably have better luck checking out other soundtracks.
As for an actual answer regarding what genre the music of Ruby and Sapphire is, I'd say, very broadly, classical (that is, lowercase C classical. Classical with a capital C refers to a specific era of music). Wikipedia states that classical music is characterized "by formality and complexity in its musical form and harmonic organization, particularly with the use of polyphony." While these are very broad terms, the music of these games (and, I'd wager, every Pokemon game and very nearly every Nintendo soundtrack) exhibit all of these characteristics, save for complex musical form, which is likely due to limitations in song length and the necessity of the tracks looping. On top of this, the composers of the Pokemon games have stated that some of their biggest inspirations are classical composers (which I discuss, among other things, in this answer. Give it a read if you want more information). That being said, music from the Pokemon games very obviously takes inspiration from many sources. Many tracks (namely the higher energy trainer battle themes) incorporate elements of popular music, for example most of the battle themes have a snare drum backbeat (a snare drum hit on beats two and four of a standard 4/4 measure), a staple of rock, pop, metal, funk, jazz, and pretty much everything else, but not commonly featured in classical music.
For the third and final act of my already too long answer, I'd like you to question with me what genre is and why it matters (pointless fluff? Likely. Let me have this, I went to college for music and dammit I'm going to use my degree). For soundtrack music especially, I think the concept of genre is incredibly flimsy and nigh impossible to define well. Check out the original theme for Mt. Chimney. Like much of the original R/S/E soundtrack, dominated by horns (specifically brass instruments - trumpets, tubas, trombones), and, because of this has a more "classical" sound to it. Now compare it to the remake from OR/AS. Funky, wah-wah guitars reminiscent of 80s action movies, and funky saxophones. Same song, obviously, but I don't think it'd be right to define them as being in the same genre. Genre is almost exclusively useful for marketing, (and, I suppose, so you can find similar music... but that's exactly why genre is a thing in marketing) and it fails really, really badly when it comes to defining certain types of music and neatly fitting them into specific parameters, and in my opinion classical music (which a good chunk of soundtracks fall under) falls squarely into that category. The music for Pokemon soundtracks can't really be put into a genre, because it's so varied. You'd have an easier time putting specific pieces into certain genres, but even that might be hard. The Lance/Red battle theme from HG/SS is effectively just a rock instrumental, but one of the main features of the piece is strings, which are front and center for a good chunk of the song, while many of the more overtly rock concepts sit in the background. Is it classical music, because of the strings and polyphony? Is it orchestral rock? Does it matter? Not really. In my personal opinion, it's not very important to put music into specific genres. Yes, it can be useful, but few people write music thinking, "I want to make [genre]!" A couple of my favorite real-world examples of this: the members of Black Sabbath (the guys who turned heavy metal into what it is now, arguably the first metal band) had nothing to do with the term "heavy metal" and initially didn't like it, and Primus' music is so weird and unique that it has its own genre tag (wanna try to define what the hell this genre is?). This is applicable to almost all music if you think about it. Can you tell me what pop music really is? Michael Jackson sounds nothing like Taylor Swift! And don't even get me started on progressive rock.
The point of this entire paragraph is to say that, while genre can be useful for finding other similar music, it will often fall short of that, and that some types of music (relevant in this instance because the soundtracks for Pokemon games are a part of this) are either so poorly defined, so specific, or so varied that genre isn't really useful. Regarding the statement in the question, if you're looking to find more music similar to the soundtrack of R/S/E, your best bet would probably be checking out the soundtracks of other Pokemon games, or perhaps Nintendo games at large. Specific to R/S/E, the constant presence of horns reminds me of march music, so it may be worth your while to check out composers of marches. Because I'm basic, I think a good place to start would be Sousa's Stars and Stripes Forever.
TL;DR: Chiptune is more of an instrument than a genre. It's hard to define exactly what genre the music of R/S/E, as well as other Pokemon games, is, but they take a lot of notes from classical music. But genre doesn't even matter because it's was invented by businesses or something and I'm a fun hating music nerd and know big words, blah blah blah something long winded about rock or metal or something.
