Due to regulations, we want to create our token (finally) to be usable in our system but not tradeable. Is this possible with a polygon chain?

For the transfer limitation, you have to implement custom logic into the before/after token transfer function according to your requirements. For example, if you want only the KYC wallet address to send and receive your token, then you can create a mapping to specify whether the address is eligible for transfer or not. And the same for the receiver address. This is just a simple example, these custom logics totally depend on your requirements and which type of condition and limitation you want to put on token transfer.