If Haaland stays longer at Manchester City I'm sure he can achieve all categories, this season Haaland has the opportunity to get a new record as the top scorer of the Champions League so far he has collected 12 goals and has no competitors.
I think that there is no need for Haaland to stay at Manchester City to go for new records. He can achieve it whatever top team he goes to actually. When I first saw his transfer from Dortmund to Manchester City I was wondering whether he would be able to adapt quickly. And he made a lightning start there. I think that we can see the same effort from him if he goes to for example Real Madrid.
There is a possibility for Haaland to leave Manchester City after one more season as well. Because it was said that there was a secret option on his contract which gets active by 2024. I assume he will have a release clause then and he seems okay to look for new adventures after playing 2-3 seasons here. But even if he doesn't stay here for a long time he would surely be remembered with big respect by Manchester City.