When I encountered him I didn’t care to take a picture until the next day when I got/was given another chance and he was [...]
I got a gift from my mother
I was given a gift by my mother.
This is much the same in English as in other languages. They give something to you; you receive something from them. Both describe more or less the same action, you just swap the subject and object of the sentence.
In this case there is a difference in nuance between "got the chance" and "was given the chance". "Got" can mean "received" but it can also imply simple opportunity, as in "I saw him on the street and took his picture (with or without his consent)". "Was given" strongly implies consent (although in some contexts it can refer to "divine providence", i.e. "God gave me the opportunity").
So if you want to be clear that he agreed to have his picture taken, you would use "was given". If you actually took the picture without consent you would say "got". If this isn't important, either will do.
I’m no English language expert, but when I hear, “we got given a new dog,” it sounds so incredibly ignorant. “We were given a new dog” sounds correct to me, or “we received a new dog.” I live in California now, coming from Onlahoma for 30 years, and I’ve heard TONS of people use the term “we got given xyz” and it’s like nails on a chalkboard to me! Maybe it’s a coloqial thing.