If just received second doctoral degree, how does one sign their name at the end of an email or other correspondence. Would it just stay Dr. So and So or be Dr. Dr. So and So?
The usual form in Austria would be "DDr.", which you see quite often, especially on lawyer's plates. After that, it would be DDDr. (and probably so on).
Remark : He is at least DDDDDDr.: https://www.nachrichten.at/oberoesterreich/Das-ist-Doktor-Doktor-Doktor-Doktor-Doktor-Doktor-Norbert-Heinel;art4,843952
Depending on the country and local department customs, signing Dr. So and So would be already ridiculous enough. But I don't know of any situation where signing Dr. Dr. So and So wouldn't come out as a show off. I particularly would think very poorly of someone signing their name in this way.
In Germany I have seen the title dres. (for doctores) used for multiple PhDs.
For real life examples, search for "Kieferorthopäde" (that is, orthodontist) and "Dr. Dr." or for "Kieferorthopäde" and "DDr.". In Germany and Austria, orthodontists typically study both medicine and dentistry, and some of them do a doctorate in both disciplines. Germans seem to prefer "Dr. Dr.", whereas "DDr" occurs primarily in Austria.
(Other countries may have other customs, but this has been my experience)
In most circumstances, no one bothers writing "Dr." as a prefix anyway! Informal situations in general tend to drop prefixes and suffixes, especially honorifics and degrees. I guarantee you that Paul McCartney doesn't make dinner reservations as "Sir Paul McCartney". Most people with doctorate degrees don't introduce themselves as "Doctor So-and-so", similarly to how people don't think of themselves as "Bob Bobrick Junior". In formal situations, this is obviously different, but writing a letter to my friend, I wouldn't sign as "The Honorable user45266" if I were a courtroom judge.
Assuming that one is in a situation where convention would dictate a person with a doctorate to sign with the title, a person with two such degrees should probably just sign as "Dr. So-and-so". If mentioning both degrees should happen to be importance, "So-and-so, Doctor of ________ and _______" should suffice.
Same situation, I am about to earn a second doctorate. I would write only Dr. FirstName LastName, or when necessary to explain my fields more specific, after LastName. But not on both ends.