Data Line Protection.
If I understand correctly, you are connecting an Ethernet cable into the UPS LAN[IN] port and then continuing the connection to your network via some USP LAN[OUT] port. I could not make out the port details and wording via the link you provide. UPS seems to be discontinued. Specs tab and "More Details" did not open any new web pages.
That said you probably have some protection for the Ethernet line but I am not sure just how much real protection is available. Joules.
The telephone line may likewise be protected using similar ports on the UPS. Telephone/ DSL IN and OUT.
The design of the UPS should route any surges (within specifications) to ground/earth so you should not need to add any special or extra ground connections. In fact doing that could be detrimental by setting up what is known as a ground loop. Or otherwise provide another undesirable path for current to flow through. Maybe the wrong way....
Where the UPS provided protection can fail is if the UPS itself is not properly grounded. Or has taken many surge hits and degraded as a result.
So my answer is yes, you do have some protection but how much, how good, and for how long are valid questions.
Here is a link that provides a good, general overview of surge protection. You can easily find similar links and explanations. Just use search criteria (words and phrasing) that matches your interest and concerns.
https://brennan-electric.com/how-do-surge-protectors-work/
Fortunately, my area does not have power/surge problems. However, if there are incoming storms or if some street work is being done then I power down and unplug for the duration. Including the UPS. No point in having the UPS absorb hits when there is nothing connected to it.