CLN: Remove StringMixin from PandasObject
- [x] closes #26495
- [ ] tests added / passed
- [x] passes `git diff upstream/master -u -- "*.py" | flake8 --diff`
- [x] whatsnew entry

Addresses #26495. In addition to the use of StringMixin in PandasObject, StringMixin is also used in core.computation and the io.pytables. I'd like to tackle those in a different PR, as there is some funkyness there (calls that depend on frame depth etc.)

# [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/26505?src=pr&el=h1) Report
> Merging [#26505](https://codecov.io/gh/pandas-dev/pandas/pull/26505?src=pr&el=desc) into [master](https://codecov.io/gh/pandas-dev/pandas/commit/65eac80dd4113ddbcee876029bfe03b4b3f11a3d?src=pr&el=desc) will **increase** coverage by `<.01%`.
> The diff coverage is `100%`.

[![Impacted file tree graph](https://codecov.io/gh/pandas-dev/pandas/pull/26505/graphs/tree.svg?width=650&token=eZ4WkYLtcO&height=150&src=pr)](https://codecov.io/gh/pandas-dev/pandas/pull/26505?src=pr&el=tree)

```diff
@@            Coverage Diff             @@
##           master   #26505      +/-   ##
==========================================
+ Coverage   91.75%   91.75%   +<.01%     
==========================================
  Files         174      174              
  Lines       50761    50676      -85     
==========================================
- Hits        46575    46499      -76     
+ Misses       4186     4177       -9
```

| Flag | Coverage Δ | |
|---|---|---|
| #multiple | `90.26% <100%> (ø)` | :arrow_up: |
| #single | `41.72% <100%> (-0.08%)` | :arrow_down: |

| [Impacted Files](https://codecov.io/gh/pandas-dev/pandas/pull/26505?src=pr&el=tree) | Coverage Δ | |
|---|---|---|
| [pandas/core/internals/managers.py](https://codecov.io/gh/pandas-dev/pandas/pull/26505/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvaW50ZXJuYWxzL21hbmFnZXJzLnB5) | `93.87% <ø> (-0.06%)` | :arrow_down: |
| [pandas/core/indexes/frozen.py](https://codecov.io/gh/pandas-dev/pandas/pull/26505/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvaW5kZXhlcy9mcm96ZW4ucHk=) | `91.78% <ø> (-0.33%)` | :arrow_down: |
| [pandas/core/series.py](https://codecov.io/gh/pandas-dev/pandas/pull/26505/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvc2VyaWVzLnB5) | `93.61% <ø> (-0.06%)` | :arrow_down: |
| [pandas/core/panel.py](https://codecov.io/gh/pandas-dev/pandas/pull/26505/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvcGFuZWwucHk=) | `17.61% <ø> (+0.11%)` | :arrow_up: |
| [pandas/core/frame.py](https://codecov.io/gh/pandas-dev/pandas/pull/26505/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvZnJhbWUucHk=) | `97% <ø> (-0.14%)` | :arrow_down: |
| [pandas/core/generic.py](https://codecov.io/gh/pandas-dev/pandas/pull/26505/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvZ2VuZXJpYy5weQ==) | `93.57% <ø> (+0.08%)` | :arrow_up: |
| [pandas/core/window.py](https://codecov.io/gh/pandas-dev/pandas/pull/26505/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvd2luZG93LnB5) | `96.4% <ø> (-0.02%)` | :arrow_down: |
| [pandas/core/internals/blocks.py](https://codecov.io/gh/pandas-dev/pandas/pull/26505/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvaW50ZXJuYWxzL2Jsb2Nrcy5weQ==) | `94.12% <ø> (+0.04%)` | :arrow_up: |
| [pandas/core/groupby/groupby.py](https://codecov.io/gh/pandas-dev/pandas/pull/26505/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvZ3JvdXBieS9ncm91cGJ5LnB5) | `97.23% <ø> (-0.01%)` | :arrow_down: |
| [pandas/core/arrays/categorical.py](https://codecov.io/gh/pandas-dev/pandas/pull/26505/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvYXJyYXlzL2NhdGVnb3JpY2FsLnB5) | `95.91% <ø> (-0.05%)` | :arrow_down: |
| ... and [16 more](https://codecov.io/gh/pandas-dev/pandas/pull/26505/diff?src=pr&el=tree-more) | |

------

[Continue to review full report at Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/26505?src=pr&el=continue).
> **Legend** - [Click here to learn more](https://docs.codecov.io/docs/codecov-delta)
> `Δ = absolute <relative> (impact)`, `ø = not affected`, `? = missing data`
> Powered by [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/26505?src=pr&el=footer). Last update [65eac80...a17e570](https://codecov.io/gh/pandas-dev/pandas/pull/26505?src=pr&el=lastupdated). Read the [comment docs](https://docs.codecov.io/docs/pull-request-comments).
 # [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/26505?src=pr&el=h1) Report
> Merging [#26505](https://codecov.io/gh/pandas-dev/pandas/pull/26505?src=pr&el=desc) into [master](https://codecov.io/gh/pandas-dev/pandas/commit/65eac80dd4113ddbcee876029bfe03b4b3f11a3d?src=pr&el=desc) will **increase** coverage by `<.01%`.
> The diff coverage is `100%`.

[![Impacted file tree graph](https://codecov.io/gh/pandas-dev/pandas/pull/26505/graphs/tree.svg?width=650&token=eZ4WkYLtcO&height=150&src=pr)](https://codecov.io/gh/pandas-dev/pandas/pull/26505?src=pr&el=tree)

```diff
@@            Coverage Diff             @@
##           master   #26505      +/-   ##
==========================================
+ Coverage   91.75%   91.75%   +<.01%     
==========================================
  Files         174      174              
  Lines       50761    50676      -85     
==========================================
- Hits        46575    46499      -76     
+ Misses       4186     4177       -9
```

| Flag | Coverage Δ | |
|---|---|---|
| #multiple | `90.26% <100%> (ø)` | :arrow_up: |
| #single | `41.72% <100%> (-0.08%)` | :arrow_down: |

| [Impacted Files](https://codecov.io/gh/pandas-dev/pandas/pull/26505?src=pr&el=tree) | Coverage Δ | |
|---|---|---|
| [pandas/core/internals/managers.py](https://codecov.io/gh/pandas-dev/pandas/pull/26505/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvaW50ZXJuYWxzL21hbmFnZXJzLnB5) | `93.87% <ø> (-0.06%)` | :arrow_down: |
| [pandas/core/indexes/frozen.py](https://codecov.io/gh/pandas-dev/pandas/pull/26505/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvaW5kZXhlcy9mcm96ZW4ucHk=) | `91.78% <ø> (-0.33%)` | :arrow_down: |
| [pandas/core/series.py](https://codecov.io/gh/pandas-dev/pandas/pull/26505/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvc2VyaWVzLnB5) | `93.61% <ø> (-0.06%)` | :arrow_down: |
| [pandas/core/panel.py](https://codecov.io/gh/pandas-dev/pandas/pull/26505/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvcGFuZWwucHk=) | `17.61% <ø> (+0.11%)` | :arrow_up: |
| [pandas/core/frame.py](https://codecov.io/gh/pandas-dev/pandas/pull/26505/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvZnJhbWUucHk=) | `97% <ø> (-0.14%)` | :arrow_down: |
| [pandas/core/generic.py](https://codecov.io/gh/pandas-dev/pandas/pull/26505/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvZ2VuZXJpYy5weQ==) | `93.57% <ø> (+0.08%)` | :arrow_up: |
| [pandas/core/window.py](https://codecov.io/gh/pandas-dev/pandas/pull/26505/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvd2luZG93LnB5) | `96.4% <ø> (-0.02%)` | :arrow_down: |
| [pandas/core/internals/blocks.py](https://codecov.io/gh/pandas-dev/pandas/pull/26505/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvaW50ZXJuYWxzL2Jsb2Nrcy5weQ==) | `94.12% <ø> (+0.04%)` | :arrow_up: |
| [pandas/core/groupby/groupby.py](https://codecov.io/gh/pandas-dev/pandas/pull/26505/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvZ3JvdXBieS9ncm91cGJ5LnB5) | `97.23% <ø> (-0.01%)` | :arrow_down: |
| [pandas/core/arrays/categorical.py](https://codecov.io/gh/pandas-dev/pandas/pull/26505/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvYXJyYXlzL2NhdGVnb3JpY2FsLnB5) | `95.91% <ø> (-0.05%)` | :arrow_down: |
| ... and [16 more](https://codecov.io/gh/pandas-dev/pandas/pull/26505/diff?src=pr&el=tree-more) | |

------

[Continue to review full report at Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/26505?src=pr&el=continue).
> **Legend** - [Click here to learn more](https://docs.codecov.io/docs/codecov-delta)
> `Δ = absolute <relative> (impact)`, `ø = not affected`, `? = missing data`
> Powered by [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/26505?src=pr&el=footer). Last update [65eac80...a17e570](https://codecov.io/gh/pandas-dev/pandas/pull/26505?src=pr&el=lastupdated). Read the [comment docs](https://docs.codecov.io/docs/pull-request-comments).
 thanks @topper-123 