I think our colleagues gave you really good answers and I wanted to add a little bit to them, even if it sounds a little exoteric or too outside the box. All this comes from my personal experience while learning to play the piano.
At some point of my journey I understood that the most important thing is the music.
The piano is just an instrument through which we will make music happen. This being said, it is also important to understand the difference between what you play to improve your mechanical technique and what you play because you love. It is like sports. If you wanna be good at it, only practicing the sport itself will not make you a pro. There's all sorts of exercises to improve your abilities and help you becoming a pro. But to me it always has to start and finish on the most important point, the love for the sport. To play it because you love it.
In other words: it is absolutely a must to play the songs you like. Of course, if they are too difficult to your level (I liked Chopin etudes...), you may either find simplified versions that may help you building your technique while maintaining your motivation or find other songs that you also like but are not that difficult. Your teacher should be able to present you these simplified versions.
The reason why I insist on loving what you are doing is because it is too easy to loose motivation and if that happens, you will not make progress and frustration will knock your door. Maybe you will even give up on learning the instrument.
Also on the subject of the piano as an instrument/medium to make music: realizing this may help you understand that there are a lot of musical knowledge that we can acquire away from the musical instrument. The basics of music theory are not a lot nor difficult to learn. The secret that a lot of people miss is while learning music theory, to link it to ear training. Why? After you learn about scales and chord progressions, if you can identify them by ear, you can understand what is going on on a lot of pop songs just by hearing them. Sounds like a super power right? But is quite doable and it is very cool. Imagine you can learn a song just by hearing it. Then the other part of the problem would be how to play that song. That is a mechanical problem, between your hands and the piano.
Long story short. If you disassemble the meaning of what is to learn the piano, you may learn it faster than other people, by making sense of what you are doing, instead of just doing what someone is telling you to do. Ballsy, I know. That's me, that's how I learned after having lessons with many classical and jazz teachers. And I don't regret it.
Addendum: on the mechanical side, I think our colleagues covered it. I would just add - no matter what part of the music you are trying assimilate, the rhythm, the notes, always keep in mind 'making music'. If you play a 5 notes melody and it sounds pleasant to listen to, then you are on the right track. Oh, how to know that? record yourself playing. Compare to the original recording. Perceiving those nuances is part of being a musician.