The application of piss or pissed to anger was first documented as an expression just after World War II. In two articles from American Speech in the same year, Fred Eikel, Jr. and Joseph W. Bishop, Jr. each documented the usage.
Here is Eikel, Fred. “An Aggie Vocabulary of Slang.” American Speech, vol. 21, no. 1, 1946, pp. 29–36. JSTOR, www.jstor.org/stable/487347, p. 33.
HE PISSED (or PEED) ME OFF. An expression used of a person who in any way disappointed the speaker.
This usage comes from Texas A&M, which at the time had a major military presence.
Here is Bishop, Joseph W. “American Army Speech in the European Theater.” American Speech, vol. 21, no. 4, 1946, pp. 241–252. JSTOR, www.jstor.org/stable/487320. p.249.
a. Pissed-off (or P'd off). This means, roughly, fed-up, irritated, depressed. I have no idea of its history. The British say browned off and it may be that the Americans who borrowed the phrase simply felt that 'browned' was not strong enough. The superlative is, for some reason, highly pissed off, which may also be a Briticism.
Bishop is an amateur recording slang from his time in service. His note - "I have no idea of its history" - carries over to explaining why this usage appeared, or what its precise relation to micturation is. The guess of an association with "brown off" is possible, as is an attraction to the vulgarity of "piss" or an affinity for another figurative usage of "piss."