The hand operated cobler sewing machine that I bought off of Amazon came with a black monofilament thread.
Same material as fishing line, just heavier than what we typically use for freshwater fishing around here. The suture kits that doctors use have a monofilament line in them in a lot of cases, probably depending upon whether the suture will be internal or external. I had one of my dogs spayed recently and when I removed the sutures, they were monofilament and had a slight tent to them. Which is good since my old eyes would have had trouble seeing them otherwise. I've also used monofilament to sew up injuries on dogs that I've owned over the years when they were injured from dominance disputes with another dog.
Although the monofilament line works for sewing leather, I don't particularly like the look of it and will buy some heavy cloth type thread when I get a chance. Something at least as thick as is used on jeans.
For sewing straps though, I don't like monofilament thread because I just do not like the knot tying properties of it. Maybe that is a reflection on the material, or maybe it is a reflection on my knot tying ability?
Is it possible to weave the yarn / string that you are using to make the leashes back on itself to make the loop, similar to what happens when you weave the strands of a rope to make an eye splice?
If not, instead of trying to find a thread that it totally invisible, you might want to consider the opposite and go for a totally contrasting thread and make the sewing a intentional design element of the strap.