Each mortgage has an interest fixation only for a fixed period. Three months before the end of fixation, the lender has to inform the borrower that it is due to expire.
You can continue with your current bank--in which case you do not need a new mortgage. Industry data suggest that 40% of consumers do not shop around. This is not very wise. It always pays to shop around, even if only to ask your bank for a better rate. Your LTV has presumably improved a lot (due to loan repayment and property appreciation), your income is presumably is better. These factors alone should imply a lower margin than what you have been paying.
In case you switch banks, you need a new mortgage for the remaining balance. This will bring with it some notary costs, but this can be worth it.
The need for either a new mortgage or just a new fixed interest agreement that replaces the old one, is usually referred to as the need for “Anschlussfinanzierung”. If you continue with the same lender it’s called “Prolongation” and if you switch to another mortgage lender it’s called “Umschuldung”.