GlobalProtect is a lot more than just a VPN service. As such, without knowing how it has been configured, it isn't really possible to answer your question properly.
For example, since GP is able to enforce "profiles" on your PC to allow you to connect to work resources, it is entirely possible that it could enforce the use of monitoring software which could indeed track usage. I don't say that this is likely, I doubt it, but it is possible.
In addition, the answer you've given yourself certainly isn't proof. One thing that a VPN could do is called "Split Tunnelling" where some traffic may be forced to go through the VPN and some not. Of course most VPN services, when the VPN is active, will apply forced routing of most if not all traffic through the VPN at which point you may well see a different route.
You should check your browsers network settings to see if, when the VPN is turned on, you have a proxy set up to use. This again is not total proof but is a common configuration for corporate VPN's.
The other thing that can happen - and I don't know the GP product so I can't say for sure whether it can/does do this - is that a product like this can create secure "bubbles" on your PC. In that case, it is common for applications, data and network traffic for work to be wrapped up by the application and kept separate from everything else on your PC. You would normally be told this when first being set up to use the application as it is important to understand how things are kept apart. If this were the case, then your normal browsing would remain private even when the VPN is on.
As to legality. You should have been asked to sign up to some terms and conditions, possibly part of the Acceptable Use Policy. This should say what the impact is of using the VPN service with personal equipment. If not, you should request this information from your organisation as soon as possible. Organisations leave themselves open to some serious legal issues if they haven't made these things very clear.