Not necessarily. In a lot of search engines that aren't like Google, it might take quite a lot of time to enter a proper, fine-tuned query, and you may want to save its parameters.
But for a Google search, I agree that search results would probably be a better fit.
If this is a genuine thing that Swedish search forms ask (word for word), then I feel that "Do you want to save THIS search?" should also be accepted, because although it's not technically the right translation, it's what most English search forms ask, and Swedish does tend to use the definite article of "the example" (e.g. "I have pain in the hand") when it is in fact meaning THIS one. That is, Swedish tends to favour "-en", "-et" more than "den har" or "det har", whereas English favours "this search", "I have pain in MY hand".
?? OK, firstly, I'm excited about the new tree so... you guys do whatever you want, fine by me! But... um, it makes sense in English? Oh, do you mean, because it's "THE search" and not "this search" like I said? It's probably more common for forms to just have a little option that says "save search" or "save this search". But anyone searching on a Swedish page will probably speak enough to get the idea anyway.