I think this very much depends on context as to which would be the best word
1) Group of people who live in the same place like city, state or nation
Citizen would seem to be pretty good. But it does imply a political region, which may not be contiguous. So you might not say they are living in the same place. For a country, though, I think it would generally work well.
However, citizens sounds a bit pompous for a small region like a town. I’d suggest locals. As a collective, community would work well also.
And as has been pointed out, if it’s a particular region, you could use the demonym: e.g. New Yorker or Texan.
2) Group of people who are part of the same institution or organization
As Rathony said, member would work well.
But there may be a more specific word depending on the type of institution or organisation. For example 'soldiers' for people who are part of the army.