For supported HyperX headsets that included a coupon code for 2-year license activation, DTS Headphone:X can be used on Xbox One and Xbox Series consoles. The same Microsoft account that the code was activated on must be used in order to enable DTS Headphone:X, as the code activates the license for your *Microsoft account*.
Please note that the above does not apply if your product's DTS functionality is provided through an included USB adapter (eg. the Cloud II Wireless DTS), rather than an activation code. In such cases, the USB being plugged in is what provides the license to use DTS on the system the USB is plugged into, only while that USB is plugged in. Xbox typically does not support the USB adapters which would provide the DTS access.