They can do a couple different things.
Sometimes they play an important functional role in helping create a smooth surface where the cable wraps around the bottom bracket cable guide. Most or all Synapses want a piece there. It's often the case that the friction you're worried about isn't so much in the middle of the cable guide where you'd expect it to rub hardest as it is where the cable first or last touches it. If those kinds of transition areas aren't very smooth you can get issues where the cable catches additional friction in one direction but not the other. It's possible to encounter a piece that was once playing this role but got worked out of place or broken.
When you find a floating piece that doesn't appear to be doing anything or have ever done anything in terms of preventing friction, one possible explanation is that it was there as a guide for installation or re-installation. I've observed that some bikes have such pieces floating inside that would seem to be too short to have been guides, since they can't run all the way through and poke out the other side. Sometimes these make noise as they rattle around. If you can be confident that they're not doing anything to prevent rub against a frame surface or hose/housing somewhere inside the downtube, these are safe to remove.
On some bikes the liner plays a role in keeping dirt or moisture from affecting the area where the cable rubs against a guide. Usually this will be on a bike that has internal routing with an open access port in the bottom bracket area where the cables would be exposed if not for the liner. Generally speaking most bikes with this kind of internal routing layout should have a liner in those areas if at all possible, because they are often necessary for low friction.
I replace cables on a lot of Synapses (which do vary a little in their cable layout) and as with most road bikes, the main tools to do it are two pieces of liner, one long enough for the chainstay run and the other for the down tube. You can get as much fresh liner as you need by making a longitudinal slit in a piece of shift housing with a razor and then whipping it against a hard surface until it all springs apart.
When creating the new piece that will go around the BB cable guide area, it's good to think about whether it will be of benefit to make it long enough on the chainstay end that it finds some kind of hard stop there such that gravity will never cause it to slip down further if the cable is slack. Sometimes I size them that way, but other times the liner is passing through some kind of tight spot on the guide that makes that worry unrealistic, in which case it's better to keep it more minimal.