Aotan is the three blue (purple) ribbons. It is worth a flat 5 points. Extra ribbons count toward Ribbons (Tanzaku), not toward a higher Aotan score.

June · Blue Ribbon

September · Blue Ribbon

October · Blue Ribbon
Like Akatan, Aotan is a flat 5 points. Those ribbons also build toward Tanzaku. Completing both Akatan and Aotan scores both sets (10 points total from the two yaku), with no extra named combo beyond that in this game.
June and October are also the months of the Butterflies and the Deer (Ino-Shika-Chō), and September holds the Sake Cup. So the Aotan months are full of high-value cards, and a hand rich in them can aim at several yaku at once.
Aotan's three months overlap heavy hitters: Butterflies and Deer for Ino-Shika-Chō, and the Sake Cup for viewing yaku and Animals. A September or October capture can therefore serve more than one plan. When you hold two blue ribbons, protect the third month aggressively — one blocked ribbon ends Aotan for the round. Completing Aotan together with Akatan is simply 5 + 5 in this game; extras beyond the six special ribbons score on Ribbons (Tanzaku), not as a named Akatan–Aotan bonus.
Aotan uses the same Nintendo-style Koi-Koi scoring on Hanafuda Legends whether you play ranked, casual, private rooms, or vs computer. Set yaku replace smaller versions of the same family (you do not score Sankō and Shikō together), while unrelated yaku such as ribbons, animals and chaff still add to your total when you stop.
If you are new to the combination, practise in the tutorial or against the computer first, then take it into online play. How to Play covers the turn structure, matching, and when to call Koi-Koi versus stopping for points.