The fact that if it was somebody else's bike she likes riding, that relatively odd and unexpected fact would have to be indicated. Barring that, it's hers.
To zero in on the English angle: riding...bike in some form is probably always going to be a valid way of translating кататься на велосипеде, but we don't just say "riding bike." It could be "riding a bike" or "riding the bike," but frankly those both sound at least a twinge odd in this sentence (the second one sounds pretty off to me except in limited circumstances). What we're left with is "riding bikes" or "riding her bike," neither of which is a particularly close match on the word-by-word level but which represent the sense of the Russian more naturally than the superficially closer versions do.
I disagree that "a" or "the" sound odd, given that we (like all Duolingo sentences) lack the surrounding context. If, for example, you are discussing transportation choices, it would be just as natural to say "My sister likes riding a bike" as "... her bike". Likewise, if you are talking about a bike you don't like, someone might respond, "My sister likes riding the bike"; granted, that last one is more of a stretch, but the point is, as far as I can tell, all three sentences would be translated into Russian in the same way, so it's pretty unclear why only one is considered valid to translate back into English.
I disagree that it would be "just as natural" to say "likes riding a bike" as "likes riding her bike." Google results for "she likes riding a bike" vs. "she likes riding her bike" bear this out. The former seem to be preponderantly either references to a single sentence in a Dora the Explorer story or in texts written by non-native speakers (many Slavic, probably translating the equivalent phrase we see here word-for-word into English) as apparent composition practice.
Of course, none of this is an argument that "likes riding a bike" shouldn't be accepted here. Is it not? I would have suspected it probably already is.
Regarding "she likes riding the bike," the "limited circumstances" I referred to were cases like the one you raise. I'd defer to a native speaker for sure, but since "кататься на велосипеде" is a fixed idiom that just means "cycling," if you really wanted to indicate that it's a certain bike she likes riding, I think you'd want to include a determiner of some sort, probably "этом."
Shady, i think the suggested translation is really a missed opportunity here.
"My sister like riding her/a/the bicycle" is the literal translation.
This translation would give learners the opportunity to learn the meaning of кататься at the same time instead of just dropping it.