There are really a couple different types of cling peaches. Most early peaches are cling (or semi-cling, or semi-free). These are somewhat different than what is considered a California cling, which is considered a true canning peach.
California cling peaches are used in the peach canning industry for a couple reasons. First those type of peaches are firmer. They are non-melting peaches (like some of the flat peaches (i.e. TangOs). This produces a much better product than cling (i.e. early) peaches most people grow here. The early cling peaches we grow here are all melting flesh peaches. Great for fresh eating, not so great for canning.
Freestone peaches are also in this category. They are great for fresh eating, but break down pretty badly, when subjected to the heat of canning (or the cold of freezing, for people who freeze peaches). They get mushy during the canning process. Everyone likes free stones for home canning, but that’s only because they are easy and fast to work up, not because they produce a superior canning product.
The second advantage of California cling peaches (for canning) is that they have no red in the flesh (not even near the pit). This is also different than the early cling peaches most people grow, which do have some red in the flesh. Red in the flesh is a disadvantage for canning because it will turn brown (oxidize) faster in the can. One exception to this is Madison, which has some red in the flesh, but supposedly doesn’t oxidize any faster. Madison is a freestone peach, popular with home canners.
Another advantage of California clings (for canning plants, that is) is that there are less shattered pits on the canning line. Automated machines are able to separate the cling peaches from the stone without any problem, but would shatter more pits of other types of peaches.
I planted a couple California clings just to see if any of my canning customers wanted to try a little better quality canning peach. As I recall, I planted Vinegold and Babygold #5. Many California cling peaches are ill suited for the Midwest, but these varieties seemed to be the best suited from what I read.
I’ve read California cling peaches are also very tasty (and sweet) for fresh eating, just firmer and smaller.
I’m guessing the cling varieties Scott mentioned above were once California cling type peaches (i.e. processing type peaches).