A quick overview. The question is about Standard French and Standard German:
In most other languages, the "u" is either "oo" or "uh" (as in "burden"). The "r" is that one that the French are so infamously bad at getting right, often defaulting to LW (veLWee difficult to get LWight indeed), or alternatively, it is a rolled "r" like in Italy, the stereotypical Scottish accent, and admittedly most Southern German accents.
The letters "r" and "u"/"ü" prompted my initial question, but after writing it out, I realise it goes much deeper. I can read Spanish or Italian, even Portuguese, provided the vocabulary is casual enough, but the pronunciation is so hugely different it might as well be from another planet. Latin languages sound a lot like each other, at least superficially, except for French with sounds totally different.
Meanwhile, French and German are completely separate, and yet, in almost every aspect, except arguably the French nasals like -on, there is nothing in French a German speaker couldn't pronounce with absolute ease, and vice-versa.
Why? Did they get these sounds (r/ü) from each other? If so, which came first and how did that happen? Why are they absent (or at least rare and unusual) in other major languages? Why are French and German so similar in pronunciation while much closer languages like Spanish or Italian sound nothing like French?
In French, the "u" is the same as the German "ü".
This is not exclusive to French and German. It is also present in Romansch and many Gallo-Italic (ie. most Italian dialects from north of the La Spezia-Rimini line) dialects, as well as pretty much every Continental Germanic language I can think of.
The "r" is pronunced the same in both languages.
Again, this is also seen in Danish, Scanian and some dialects of Norwegian and Portuguese. It's also a relatively recent development in Standard French and German, and is not present in Austro-Bavarian/Swiss Dialects or in many traditional rural Metropolitan French or Quebecois dialects.
often defaulting to LW
I have never, ever heard a French person do this.
there is nothing in French a German speaker couldn't pronounce with absolute ease, and vice-versa.
I disagree, the 'h' sound, as well as the 'ch' sounds in the German words Ach and Ich respectively, are very hard for French people to pronounce. And German people speaking French sound just as terrible as English people speaking French.
Why are they absent (or at least rare and unusual) in other major languages?
They aren't particularly. I don't know how either /y/ or /R/ developed originally, I imagine there is some element of Sprachbund, but I also wouldn't be terribly surprised if they developed in both languages independently.
Why are French and German so similar in pronunciation
French pronunciation had a lot of Germanic (as well as indigenous Celtic) influence in its early history, but nonetheless I wouldn't go as far as to say their pronunciation is particularly similar.
This answer might not be "detailed" enough, but French is my first language, and I've spent a long time with German exchange students.
While they might sounds similar to an outsider (and again, you're the first person I've seen mention a similarity between French and German. Catalan and French are much more similar in my opinion, as well as german and dutch) to a native speaker German sounds nothing like French.
As I said, I am not a linguist, but German pronunciation is more guttural (from the throat? as I said, NMFL).
I believe you're also mistaken on other pronunciations: Italians do not roll their "r", like you said.
I hope someone can provide a nore complete answer.