What is this symbol called which is used to denote "II" in the second season of "The Quintessential Quintuplets"?
No, you're right, that is absolutely the double integral sign. There's a trend of indicating follow-on seasons of shows with indicators like random punctuation marks or letters (see, e.g. this question about Nisekoi: or this one about Dragon Ball Z), and this is almost normal by comparison since it does actually look a bit like a stylised II rather than, for example, adding an apostrophe or a random letter.
It's also vaguely relevant to the show itself - since Uesugi was brought on to tutor the five girls in subjects such as mathematics, which potentially includes a little bit of calculus.