Incorrect title?

Hi all, quick question.

I got a new job about 7 months ago and pretty much the job that I interviewed for is not the job that I am doing. First week in they basically changed what I was doing and after doing it for so long I actually enjoy it and want to grow in that direction.

I already asked about changing it and they said they “want to keep me in this hybrid position” which is not ideal for me but long term what could happen?

Is having the incorrect title going to hurt me in the future? Should I push to get my title changed? Should I start looking for other jobs that actually have the work that I want to do?

Any and all advice would be greatly appreciated