Bias tape is generally used for curves, but quilters will bind quilts with bias tape too because the theory is that the edges will wear better since there are more threads around them. Straight of grain tape means the same few threads are the ones that will get rubbed, and possibly wear through more quickly. But it's just generally all called bias tape, and the makers are called that because same reason. What I have found is that both seem to work equally well. I have never had a quilt's binding wear through, perhaps because none of my quilts are long enough to hang to the floor and I don't drag them around. I quit doing bias unless I prefer the look of the fabric on the bias. Strips usually look better angled, for one.
For the masks, the group that I donated through is using strips of t-shirts. They don't fray, and one good tug makes them curl in on themselves so there's no need to sew. They are roundish, but as long as the casing isn't too big, they will hold their place nicely.
I just finished making straight of grain ties with my bias tape maker, and they're really snug in the casings. As long as the knot doesn't come out and I have placed it correctly on my face, I don't think I will need to worry that it will slip. That's the main concern, that people will fiddle with them, thus contaminating their hands and spreading things around, or their hands are unknowingly contaminated and they fiddle with the mask and so infect themselves.