That is a very good question.
One answer to your question is that it is difficult to learn piano from Synthesia. Synthesia is a practice tool, not a tutorial. The best way to learn is from a teacher who can see what you are doing and give you exercises and explanations tailored to your needs. If that is not possible, a self-study course that provides explanations along with music can be used. Synthesia by itself will leave you with many questions and allow you to develop bad habits.
But to try to answer your question, first you have to understand that the piece you are playing is a simplification of Für Elise arranged by Gilbert DeBenedetti to give a beginner the feeling of playing a piano classic. Next you have to understand that piano fingering is not absolute; it can lead to sometimes furious debates about how it is best done. My guess is the fingering for Für Elise should be chosen to emphasize the phrasing of the piece. (But I have never studied Für Elise with a piano teacher.) Groups of notes that are played with the same hand tend to flow together more easily than those that are split between the hands. Sometimes one chooses a less obvious fingering that produces a better sound.
That said, I don't think the fingering shown by Synthesia is correct. This is a link to the sheet music and videos from DeBenedetti's G Major Music Theory website:
As is typical of beginner pieces, DeBenedetti has arranged the piece so that both hands play in a fixed 5 finger position. Right hand thumb is on middle C and the remaining fingers play notes that naturally fall under the finger. You do have to change from white to black keys with the same finger. Left hand thumb is on G below middle C and the remaining fingers are on the notes naturally under the fingers. I disagree with DeBenedetti's decision to write the long notes as dotted half notes. I think writing them as quarter notes would be a better reflection of the original.
So you are correct that DeBenedetti intended for you to play those repeated notes with your right hand, not the left as shown by Synthesia. But when you get to the real Für Elise, which won't be for some time, the fingering will have little resemblance to what you are doing now. If you weren't a beginner, a teacher might have you play blue notes 4-6 in your screenshot with the right hand to get more of the feel of the real piece with 3 note phrases in the left hand answered by 4 note phrases in the right hand. This shows the phrasing I am suggesting:
But for now, play the first two green eighth notes with your left hand and the third eighth note with your right. Do shorten those long notes to quarter notes so you can restrike the note as an eighth as part of the four note phrase.