Does anyone know the background or history of these songs (if any)? Seems like German but I don't know if they have any history in operas or such.

They were used in a Korean TV drama series (The K2).

Anemone
https://youtu.be/xweJE3eOPng

Serenade
https://www.youtube.com/watch?v=fPMM-wwwmWk

I believe the first one is Anemone by ClariS (Female version).

"ClariS released their 12th single "Anemone" (アネモネ) on July 29, 2015; the song is used as the ending theme to the 2015 anime series Classroom Crisis."

-Wiki

The second one is Ständchen ("Serenade") by Franz Schubert.

"Schubert's immortal "Serenade" was written in 1826. it is so familiar that it needs no analysis, nor is one necessary from any point of view. It is simply a lovely melody from first note to last, written upon the inspiration of the moment, and yet characterized by absolute perfection of finish and a grace and beauty of which one never tires. It was originally composed as an alto solo and male chorus and was subsequently rearranged for female voices only. The circumstances of its composition as told by Schubert's biographer, Von Hellborn, are of more than ordinary interest. Von Hellborn says:

"One Sunday, during the summer of 1826, Schubert with several friends was returning from Potzleinsdorf to the city, and on strolling along through Wahring, he saw his friend Tieze sitting at a table in the garden of the 'Zum Biersack.' The whole party determined on a halt in their journey. Tieze had a book lying open before him, and Schubert soon began to turn over the leaves. Suddenly he stopped, and pointing to a poem, exclaimed, 'such a delicious melody has just come into my head, if I but had a sheet of music paper with me.' Herr Doppler drew a few music lines on the back of a bill of fare, and in the midst of a genuine Sunday hubbub, with fiddlers, skittle players, and waiters running about in different directions with orders, Schubert wrote that lovely song."

-Music with Ease

Thanks . But as I listen to samples online of these songs:

1) ちょっと 違う でしょうね

2) also seems different than the song in question.

The origins of the songs in question remains a mystery.

Chigau? Ah, zannen desu ne. I wasn't sure about the first one, but thought for sure it was the Schubert.

I actually have the Schubert version of Serenade but didn't know the history of it. It's just been a nice piece of music until you shed some light. So thanks.

I'm interested in the posted works though - I'd like to get similar works.