PERF: Index.join to maintain cached attributes in more cases
- [x] [Tests added and passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#writing-tests) if fixing a bug or adding a new feature
- [x] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit).
- [x] Added [type annotations](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#type-hints) to new arguments/methods/functions.
- [x] Added an entry in the latest `doc/source/whatsnew/v3.0.0.rst` file if fixing a bug or adding a new feature.


```
import pandas as pd

data = [f"i-{i:05}" for i in range(100_000)]
dtype = "string[pyarrow_numpy]"

idx1 = pd.Index(data, dtype=dtype)
idx2 = pd.Index(data[1:], dtype=dtype)

# the is_unique call at the end is cached in this PR
%timeit idx1.join(idx2, how="outer").is_unique

# 59.1 ms ± 1.29 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)  -> main
# 41.9 ms ± 894 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)   -> PR
```
Thanks @lukemanley 