Hi @mukashinoyume,
Yes, you should let your employer know and ask them to update your details.
While your tax records with us are now correct, employers use the details they hold in their payroll systems when reporting salary and wages to us through Single Touch Payroll (STP). To make sure your income info matches correctly and avoids any issues later (for example, with pre‑fill or identity checks), your employer should update your name (and sex marker, if they collect this) to match your current legal details.
Providing your employer with a new tax file number declaration or withholding declaration will give them what they need to update your details through their payroll or HR system 😊