Wait, what? Holy cats, I'm only now realizing we never know her name! The show is so intimately first-person that never even occurred to me!
Absolutely. That's for a couple of reasons, but mainly because the conceit behind The Office is that there's a documentary film crew watching. The conceit in Fleabag is much more intimate. She's looking at you, talking to you, directly, not through a camera. The Office introduced something referred to as "camera awareness", in which the characters know they're being filmed. By contrast, Fleabag is a first-person story living inside a third-person presentation.
God, no, please no second season. Fleabag doesn't feel like a series, and for good reason. Waller-Bridge wrote it -- and maybe even staged it? -- as a play. It's a feature-length self-contained story with specific reveals based on learning who the character is. Once you learn who the character is -- her relationship to Boo, her sister, her father, her dead mother -- the story's work is done. Anything additional would feel like a pointless epilogue.