There are pros and cons to being an adult student, and it cannot be compared to how children learn. Adults are more mentally mature, understand the importance of work ethic and learn piano because they want to. Not because their parents signed them up for lessons.
You want to be better than you are.
Be patient and kind to yourself because you won't be learning from the method books forever. Just because the pieces are children songs doesn't equate to them being easy. Learning piano is an alien skill, no-one was born as a baby knowing how to play. It requires understanding of how to read sheet music, playing with independent fingers and multiple fingers, playing with both hands, hand movements and technique. You are doing ok.
You are bored of uninspiring children beginner music
Method books (e.g. John Thompson's, Me and My Piano, Piano Time, Piano Adventures, Piano Safari) serve to teach the basics and build your first foundation. So don't think your teacher is patronising you by teaching you 'Old MacDonald', everyone goes through the same rite of passage. Your teacher should regularly check-in with you what pieces you want to learn, communicate your current progress and learning plan. Trial learning by rote or ear.
The piece are too hard and an uphill struggle
Often a problem for self-taught players, it can be difficult to gauge your own current ability and plan what the next steps are to progress forwards. Perhaps you dream of playing Chopin's Fantasie Impromptu by can't get past the few bars without breaking out in frustration. Mentally you might be ready in terms of appreciation, but physically you can't at that level yet. Pick easier music and learn technical exercises! A Dozen A Day, Piano Time Classics, John Thompson's First Classics are just some great books.
You don't understand the link between the sheet music and music you enjoy
It's all music, but if the style of music you enjoy listening to is different to the music played in lessons, then it's very easy to separate the two. And where's the link between what you enjoy and what you are learning? I suggest playing your radio/ playlist music on speaker, and try playing along with some of it. Whether it's just the melody or basic chords. Also find arrangements of these songs that are level appropriate for you to learn on piano.
You are lazy
Just do it, no excuses. Stop procrastinating.
Your hands just don't work together
Instinctively both our hands want to copy each other. And it takes time to split them up both in the method of reading notes and the finger pattern on the keys. If it's still a problem after 1 year, so back to basics and include finger exercises like A Dozen A Day.
Your inner voice says 'you suck!'
We are our own worst enemy. Stop being so hard on yourself, you are here to learn and be a beginner. Don't punish your mistakes, learn from them. So be kind to yourself.
You're too sensitive to criticism
When there are corrections or feedback given from the teacher, you may... make excuses, joke about your failure, feel uncomfortable, apologise, try and justify yourself. Learn to listen to criticism without seeing it as a personal attack. Teachers should be there to help you improve and have your best interests at heart.
You don't want to spend money on yourself
I'll be honest, learning a musical instrument is not a cheap hobby. You pay for the piano, lessons, books and exams. And it can be very tempting to just rely solely on the internet for your sheet music. There is nothing wrong with getting sheet music from the internet... it's just very disorganised and inconvenient. If you are learning music in a serious manner, your books will be with you for a lifetime. You deserve to invest in yourself and books will inspire you. I still have all of my books from when I was a student and still dip back into them to play old favourites.
Your close-minded about what you want to learn
Music is very wide-ranging, from jazz, blues, pop, modern, classical, romantic, world, baroque... Be open-minded and try out other genres, you might find a new favourite. Classical music may not be your favourite, but it's important for your musical education. Just like vegetables, it is good for you!
You get nervous playing for your teacher, family or friends
Nerves unfortunately can affect any student, regardless of age, experience or ability. Just remember that your audience are there to support you and see you progress. Performance skills are a separate set of skills to normal lessons or practising, and so need to be developed. This is done through doing regular performances and home recordings.