"Herd", implies natural populations when I read it. There are seasonal aggregations of the giant insular species, who also aggregate around localized resources, i.e. pools of water, but they are not what might be thought of as herd animals.
PCHP has mentioned that on some volcanoes in the Galapagos there are trodden paths up and own the mountain sides, where the size of the path is a perfect match to tortoise sizes. It could be, maybe, that at one time when there were thousands and thousands they moved on a seasonal cycle up and down these paths, and that would constitute herd movement.
There are old paintings from the first explorations of the Indian Ocean Islands where tortoises are about a field as stones in a quarry, almost shell to shell. This predates massive exploitation, and again may be what constitutes a herd.
Pancakes are nearly always found in very close proximity to each other, and it would seem the carrying capacity of where they live is determined by how much shelter is available, and how many tortoises will fit. Similar to a hibernation space for snakes. It reflects the cooperative use of a limited resource, but might not be what a herd is.
In captivity all bets are off. Crowded animals at some level of crowding stress and all natural behaviors (competition for mates, fighting, etc. ) stop. Two males are sometimes OK, if no females are present, but ad a female and those same two males will not be 'buddies' anymore.
Also, most species in the wild spread out upon hatching or soon thereafter, and don't socialize, whereas in captivity, they become socialized by simply being kept together. This can sometimes be tricky with some eating each other.
Your initial question is pretty general, and there are many situational things to consider, right down to the individual animals. For the most part tortoises are not social, they do not herd, and stress in groups when resources are limited.