In my hood, mangos are served peeled and stuck on a wooden stick - by the same street vendors who supply the ever coveted paletas and mayonnaise-slathered elotes.
They squeeze lime juice all over the mango and cover it with salt. This combination of incredibly sweet and tangy flavors with the salt makes for a truly delicious treat. There is simply no way to avoid getting strands of mango in your teeth, it is best to accept them and deal with them once you are finished eating.
Also: when starting out with a fresh mango, it is not always necessary to skin it before eating. It can be sliced with the skin still on, then the sweet orange flesh of the fruit can be removed by grinding the slices through your teeth in a way similar to how many people enjoy the individual leaves of an artichoke.