If the company was covered by PF right when you joined in 2019, then the employer cannot register you now with a date of joining as 2023. There can be one reason why you were not given PF when you joined. That is, your salary was above Rs 15,000, and you were never covered by EPF earlier. That is, you belonged to a category called an excluded employee. If that was the reason, the employer can offer PF now as a matter of investment without the Pension fund. Then for all purposes of PF, your date of joining will be 2023.
If, on the other hand, you were not an excluded employee when you joined in 2019 but due to oversight or ignorance of the law, the employer failed to register you in PF, then the employer is liable to put the date of joining as 2019 and contribute to your PF (both your share as well as his share) together with interest and damages as per the Act, which will be computed later on.
The date of joining in PF will not have any direct bearing on your future career check. It will be the service certificate that the future employer will be verifying and not the date of joining in PF. When joining a new company, it is only the UAN that you share with them and not the service history. Even if the future HR asks about the difference, you can just say that you were an excluded employee when you joined but later on in 2023, the company decided to cover all employees irrespective of their salary, and I was also given PF coverage since 2023. That will justify the date difference.