I wanted to comment, but an actual google search gave some sort of answer for me, maybe there will be better answers than mine, I see the practical side of the fastening problem. The handles are typically sort of "D" shaped where the straight side is the shield or some straps with two fixed points on the shield body having enough space between them for an arm or hand. The shape is ideal to apply independently from the shield any regular rope, leather strap to fasten it to your armour, or clothes and make it easily accessible.
Let's see some examples:
First picture is Arc d' Orange shield with handle. A strap would perfectly fit in, and could be used for fastening it to armour.
Second picture is a Viking shield replica. There you can see an applied strap for carrying/apply on the arm, but the handle here has enough space to put a rope or strap in it, so even without the applied strap, it would be easily attachable to anywhere.
If I was a dark age soldier and I would have to carry a shield without having a purposely attached fastener, I would use a rope, tie it into handle, if the shield is moving, make a triangles around the shield to make it stable, and fix it on myself, like a backpack, maybe there are better ideas, this is my first one to try. Here is the illustration I made in paint: