I assume its purpose is to act as a filter... Is this, in fact, it's purpose?
YES, but it may also be serving as a gasket. Remove it and reassemble to check for leaks, to confirm whether the rubber part (with or without the mesh) is needed.
is the mesh necessary in my case?
It depends on your water quality and plumbing type/condition. You seem to be capable of disassembling the shower head so if debris were to cause decreased performance in the future, you could just take it apart and clean out the debris.
would removing the mesh increase the... pressure out of the shower head?
What you probably are seeking is increased flow, not pressure. Removing restrictions would actually decrease the pressure. An example would be when you put your thumb over the end of your garden hose, reducing the flow; this increases the back-pressure within the hose, causing the reduced water flow from the end of the hose to come out at a higher pressure. When you remove your thumb, flow increases but pressure drops.
The flow will increase only if the filter itself is significantly contributing to the restriction. Many modern shower heads are designed specifically to reduce flow; in some locales it is mandated by government regulations. Oftentimes the primary restriction is small water passage(s) that can be drilled larger by the end user, or the actual spray nozzles are restrictions.