有 is used for the verb ある when it has to do with possession, and the stem of that verb is あり, so reading 有 as あり isn’t that weird.
(By the way, うま is a kun’yomi of 馬. The standard on’yomi is ば.)
Kanji can have a lot of pronunciation because they were taken from Chinese and slapped onto Japanese words with different pronunciations (sometimes to try to portray meaning of the word in relation to the Chinese meaning, sometimes to try to match the sound even when it wasn’t an exact sound/meaning match), and both the new Japanese pronunciation(s) and the original Chinese pronunciation can be used in various situations. (The animal “horse” is 馬 [ま] in Chinese but 馬 [うま] in Japanese. )
By the way OP you already got plenty of answers but just keep in mind that 有馬 is a name and albeit it’s a common name with a common reading, in reality in Japanese names can really by a “free for all” when it comes to readings. You could call your son something like 騎士 but give it the reading ナイト (knight) because it “sounds cool” and nobody could complain (well, your son will probably hate you).
You can look up these kinds of weird/cringy names online by looking for キラキラネーム
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有：あり
馬：ま
有 is used for the verb ある when it has to do with possession, and the stem of that verb is あり, so reading 有 as あり isn’t that weird.
(By the way, うま is a kun’yomi of 馬. The standard on’yomi is ば.)
Kanji can have a lot of pronunciation because they were taken from Chinese and slapped onto Japanese words with different pronunciations (sometimes to try to portray meaning of the word in relation to the Chinese meaning, sometimes to try to match the sound even when it wasn’t an exact sound/meaning match), and both the new Japanese pronunciation(s) and the original Chinese pronunciation can be used in various situations. (The animal “horse” is 馬 [ま] in Chinese but 馬 [うま] in Japanese. )
By the way OP you already got plenty of answers but just keep in mind that 有馬 is a name and albeit it’s a common name with a common reading, in reality in Japanese names can really by a “free for all” when it comes to readings. You could call your son something like 騎士 but give it the reading ナイト (knight) because it “sounds cool” and nobody could complain (well, your son will probably hate you).
You can look up these kinds of weird/cringy names online by looking for キラキラネーム