BUG: fix Series.str.fullmatch() and Series.str.match() with a compiled regex failing with arrow strings 
Fixes: #61952
After Fix:
```python
DATA = ["applep", "bananap", "Cherryp", "DATEp", "eGGpLANTp", "123p", "23.45p"]
s=pd.Series(DATA)
s.str.fullmatch(re.compile(r"applep"))

Output:
0     True
1    False
2    False
3    False
4    False
5    False
6    False
dtype: bool
```
```python
DATA = ["applep", "bananap", "Cherryp", "DATEp", "eGGpLANTp", "123p", "23.45p"]
sa=pd.Series(DATA, dtype="string[pyarrow]")
sa.str.match(re.compile(r"applep"))

Output:
0     True
1    False
2    False
3    False
4    False
5    False
6    False
dtype: boolean
```
- [x] closes #61952
- [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).

@jorisvandenbossche Moved tests to `pandas/tests/strings/test_find_replace.py` and made a minor change to the docstring. I’m not sure what changes need to be made in docs. could you please provide more details? > I’m not sure what changes need to be made in docs. could you please provide more details?

The suggestions of @yuanx749 are in the good direction
 Thanks @khemkaran10  Owee, I'm MrMeeseeks, Look at me.

There seem to be a conflict, please backport manually. Here are approximate instructions:

1. Checkout backport branch and update it.

```
git checkout 2.3.x
git pull
```

2. Cherry pick the first parent branch of the this PR on top of the older branch:
```
git cherry-pick -x -m1 3cefa1ee6b30843a24065fa67392fbfa63d0769b
```

3. You will likely have some merge/cherry-pick conflict here, fix them and commit:

```
git commit -am 'Backport PR #61964: BUG: fix Series.str.fullmatch() and Series.str.match() with a compiled regex failing with arrow strings '
```

4. Push to a named branch:

```
git push YOURFORK 2.3.x:auto-backport-of-pr-61964-on-2.3.x
```

5. Create a PR against branch 2.3.x, I would have named this PR:

> "Backport PR #61964 on branch 2.3.x (BUG: fix Series.str.fullmatch() and Series.str.match() with a compiled regex failing with arrow strings )"

And apply the correct labels and milestones.

Congratulations — you did some good work! Hopefully your backport PR will be tested by the continuous integration and merged soon!

Remember to remove the `Still Needs Manual Backport` label once the PR gets merged.

If these instructions are inaccurate, feel free to [suggest an improvement](https://github.com/MeeseeksBox/MeeseeksDev).
                 Manual backport -> https://github.com/pandas-dev/pandas/pull/62113