The good news is that the message is just that, a message, because according to XDA, it seems that despite being told that the feature is in offline mode, Google Assistant will continue to function as per normal. This means that you can more or less ignore the offline mode message and keep using it like you normally do.
However we guess for users who are seeing this message for the first time, it could be a bit disconcerting so hopefully this clears it up. It is unclear as to why Google Assistant is showing offline mode when used with VPNs, but it has been speculated that it could be a bug and not intentional.
XDA also notes that this seems to be the case across all devices and also regardless of how you enabled the feature. Google Assistant is currently exclusive to Pixel devices, but it has also been discovered that non-Pixel phones with the Nougat update can enable it.