I honestly don't think we have enough information to either understand the full nature of the Upside Down, including how the different matter between realities work, and/or the full nature of Shadow Monster, considering what the final scene is -- and considering we don't actually see how the tunnels are being made or how the Shadow Monster comes to exist--if it was related to the [death of the] first Demogorgon or if it is separate and pre-existed/co-existed during the events of season one??
But we might be able to presume that one reason for the underground tunnels has to do with the gateway in lab being below ground to begin with and that maybe because of the way viewers were told the Shadow Monster (and it's hive-mind) operates, being like a virus or cancer, may mean that the only way it can begin to infiltrate the characters dimension/reality is by "growing" into it???
The ending implies that we really didn't get underneath this Shadow Monster, but disrupted it's mind and sealed off the lab gate way, but there is also that portal in the woods from season one we don't understand how it came to exist either -- and, for whatever reason, it was clear that the Shadow Monster itself could not yet cross the threshold(s) between realities and was using the Demidogs, black organic vines in the tunnels, & Will to begin it's goal, which may go beyond what the characters were suggesting with the Mind Flyer analogy, since again we don't know the methodology or philosophy of "how" it wants to "dominate"---we don't know exactly to what end it wants to conquer the characters' reality and if it needs/will ever become present to do so.
However, Stranger Things JUST got an after show called Beyond Stranger Things! Perhaps there will be more insight given there and someone can give you a more solid answer, as I have not had time to watch them yet.