I wouldn't let this one grow on a tree or shrub. It's quite dense and that will shade out the support plant. I'd also cut it back ASAP after blooming before it sets seed so that it doesn't spread any more. It can be quite a problem in wild areas where it's warm enough to set seed.

I had a very well established SAC in my old garden. It was planted to grow on a large wrought iron screen acquired from an architectural salvage yard. Even with severe pruning each spring, it quickly engulfed the 6'x8' screen and started moving :-) By the end of summer, it had spread to grow along the fence behind the screen (in both directions), into and through a large burning bush and up into a very large apple tree! It was not an issue with either the burning bush or the apple, probably because the mass of the vine was focused on the screen and these were just long vining tendrils that grew into the woodies - not to any sort of smothering degree! It was rather appealing to see both of these plants with SAC blooms on them in late summer :-)) I did prune off all the tendrils once the leaves fell in fall.

But babs makes a good point - I would not consciously plant this vine to grow specifically on a woody plant (as its main support structure). It is just too large and too vigorous a grower and would swamp anything but a well established, large shade tree. I just grew a lot of clematis in a not very large garden and they sort of decided exactly where and what they were going to climb!! Good thing my apple was so large, as it got it from all sides.......a tangutica was planted intentionally to grow up into it and then it got the SAC from one side and an equally aggressive armandii from the opposite direction!! I called it my "clematis tree" :-)) And it still produced wonderful apples each year!

btw, neither the SAC nor the burning bush are invasive in my area. YMMV

SAC is an issue as far as invasiveness in southern NY. It isn't here since it's too cold to ripen seed.