Three labels with different jobs
Your Steueridentifikationsnummer is a personal identification number intended to remain with you. After first address registration in Germany, it is commonly delivered by post. Employers ask for it when setting up payroll.
A Steuernummer is issued by a tax office for a tax matter. Freelancers and businesses may receive one after submitting registration information. Its format and use differ from the personal tax ID.
Your Steuerklasse influences wage-tax withholding for employment. It is not a final calculation of every tax liability and does not replace an annual assessment where one applies.
What to retain
Keep tax-office letters, annual wage statements, health-insurance confirmations and records of income received before and after the move. Married people should retain marriage evidence in the form required for their case. Cross-border income can change the analysis, so ask a qualified tax adviser before assuming that payroll withholding settles everything.