Translation:My big brother is tall.
わたしの is unnecessary here as あに refers specifically to the speaker's older brother
It's language we are taking about here. The speaker is clearly emphasising that it's Their brother that is tall.
What Tara is saying is that 兄 only ever refers to the speakers older brother. You would never call someone else's sibling 兄/あに (or 姉/あね). Hence why the 私の is not needed.
あに refers to the speaker's social inner circle's big brother, so it is possible that あに can refer to someone else's brother.
Interestingly, height as a measurement is 身長(しんちょう). While it can be used to make comparitive statements or general observations about height, that use isn't quite as common.
When i was in Japan a few people said things like 高いですね (I'm about 6'5"), and I don't think they included 背. As a casual comment, would you say people just say 高い?
Well, it's certainly not impossible. If you're 6'5" I'd say the context is pretty unambiguous! The last time I recall someone looked up at me they included it: 背が高いですね (followed by a casual 鼻も大きい - "also a big nose", since she was speaking to my gf and didn't realize I could speak Japanese as well ^_^).
Seems a bit rude to comment on your nose being big, whether you can understand it or not...
Having a "tall" nose is a compliment (鼻が高い), having a big nose, well.....
I first learned 高い as meaning "expensive" so that could be one (admittedly odd) alternative translation! The 背(せ) thing seems weird to us but I suppose it does create some clarity.