CLN: Remove compat.re_type
- [ ] xref #25725 and #26252

This removes ``compat.re_type`` from the code base.
# [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/26279?src=pr&el=h1) Report
> Merging [#26279](https://codecov.io/gh/pandas-dev/pandas/pull/26279?src=pr&el=desc) into [master](https://codecov.io/gh/pandas-dev/pandas/commit/f46ab96fc3f6fb2e764295b4922f5f5e2323d2d8?src=pr&el=desc) will **decrease** coverage by `51.26%`.
> The diff coverage is `100%`.

[![Impacted file tree graph](https://codecov.io/gh/pandas-dev/pandas/pull/26279/graphs/tree.svg?width=650&token=eZ4WkYLtcO&height=150&src=pr)](https://codecov.io/gh/pandas-dev/pandas/pull/26279?src=pr&el=tree)

```diff
@@             Coverage Diff             @@
##           master   #26279       +/-   ##
===========================================
- Coverage   91.98%   40.72%   -51.27%     
===========================================
  Files         175      175               
  Lines       52379    52377        -2     
===========================================
- Hits        48183    21329    -26854     
- Misses       4196    31048    +26852
```

| Flag | Coverage Δ | |
|---|---|---|
| #multiple | `?` | |
| #single | `40.72% <100%> (-0.16%)` | :arrow_down: |

| [Impacted Files](https://codecov.io/gh/pandas-dev/pandas/pull/26279?src=pr&el=tree) | Coverage Δ | |
|---|---|---|
| [pandas/compat/\_\_init\_\_.py](https://codecov.io/gh/pandas-dev/pandas/pull/26279/diff?src=pr&el=tree#diff-cGFuZGFzL2NvbXBhdC9fX2luaXRfXy5weQ==) | `81.08% <ø> (-13.8%)` | :arrow_down: |
| [pandas/core/dtypes/inference.py](https://codecov.io/gh/pandas-dev/pandas/pull/26279/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvZHR5cGVzL2luZmVyZW5jZS5weQ==) | `83.6% <100%> (-16.4%)` | :arrow_down: |
| [pandas/io/formats/latex.py](https://codecov.io/gh/pandas-dev/pandas/pull/26279/diff?src=pr&el=tree#diff-cGFuZGFzL2lvL2Zvcm1hdHMvbGF0ZXgucHk=) | `0% <0%> (-100%)` | :arrow_down: |
| [pandas/io/sas/sas\_constants.py](https://codecov.io/gh/pandas-dev/pandas/pull/26279/diff?src=pr&el=tree#diff-cGFuZGFzL2lvL3Nhcy9zYXNfY29uc3RhbnRzLnB5) | `0% <0%> (-100%)` | :arrow_down: |
| [pandas/core/groupby/categorical.py](https://codecov.io/gh/pandas-dev/pandas/pull/26279/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvZ3JvdXBieS9jYXRlZ29yaWNhbC5weQ==) | `0% <0%> (-100%)` | :arrow_down: |
| [pandas/tseries/plotting.py](https://codecov.io/gh/pandas-dev/pandas/pull/26279/diff?src=pr&el=tree#diff-cGFuZGFzL3RzZXJpZXMvcGxvdHRpbmcucHk=) | `0% <0%> (-100%)` | :arrow_down: |
| [pandas/tseries/converter.py](https://codecov.io/gh/pandas-dev/pandas/pull/26279/diff?src=pr&el=tree#diff-cGFuZGFzL3RzZXJpZXMvY29udmVydGVyLnB5) | `0% <0%> (-100%)` | :arrow_down: |
| [pandas/io/formats/html.py](https://codecov.io/gh/pandas-dev/pandas/pull/26279/diff?src=pr&el=tree#diff-cGFuZGFzL2lvL2Zvcm1hdHMvaHRtbC5weQ==) | `0% <0%> (-99.37%)` | :arrow_down: |
| [pandas/io/sas/sas7bdat.py](https://codecov.io/gh/pandas-dev/pandas/pull/26279/diff?src=pr&el=tree#diff-cGFuZGFzL2lvL3Nhcy9zYXM3YmRhdC5weQ==) | `0% <0%> (-91.16%)` | :arrow_down: |
| [pandas/io/sas/sas\_xport.py](https://codecov.io/gh/pandas-dev/pandas/pull/26279/diff?src=pr&el=tree#diff-cGFuZGFzL2lvL3Nhcy9zYXNfeHBvcnQucHk=) | `0% <0%> (-90.1%)` | :arrow_down: |
| ... and [132 more](https://codecov.io/gh/pandas-dev/pandas/pull/26279/diff?src=pr&el=tree-more) | |

------

[Continue to review full report at Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/26279?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/26279?src=pr&el=footer). Last update [f46ab96...a402489](https://codecov.io/gh/pandas-dev/pandas/pull/26279?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/26279?src=pr&el=h1) Report
> Merging [#26279](https://codecov.io/gh/pandas-dev/pandas/pull/26279?src=pr&el=desc) into [master](https://codecov.io/gh/pandas-dev/pandas/commit/f46ab96fc3f6fb2e764295b4922f5f5e2323d2d8?src=pr&el=desc) will **decrease** coverage by `<.01%`.
> The diff coverage is `100%`.

[![Impacted file tree graph](https://codecov.io/gh/pandas-dev/pandas/pull/26279/graphs/tree.svg?width=650&token=eZ4WkYLtcO&height=150&src=pr)](https://codecov.io/gh/pandas-dev/pandas/pull/26279?src=pr&el=tree)

```diff
@@            Coverage Diff             @@
##           master   #26279      +/-   ##
==========================================
- Coverage   91.98%   91.98%   -0.01%     
==========================================
  Files         175      175              
  Lines       52379    52377       -2     
==========================================
- Hits        48183    48178       -5     
- Misses       4196     4199       +3
```

| Flag | Coverage Δ | |
|---|---|---|
| #multiple | `90.53% <100%> (-0.01%)` | :arrow_down: |
| #single | `40.72% <100%> (-0.16%)` | :arrow_down: |

| [Impacted Files](https://codecov.io/gh/pandas-dev/pandas/pull/26279?src=pr&el=tree) | Coverage Δ | |
|---|---|---|
| [pandas/compat/\_\_init\_\_.py](https://codecov.io/gh/pandas-dev/pandas/pull/26279/diff?src=pr&el=tree#diff-cGFuZGFzL2NvbXBhdC9fX2luaXRfXy5weQ==) | `94.59% <ø> (-0.28%)` | :arrow_down: |
| [pandas/core/dtypes/inference.py](https://codecov.io/gh/pandas-dev/pandas/pull/26279/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvZHR5cGVzL2luZmVyZW5jZS5weQ==) | `100% <100%> (ø)` | :arrow_up: |
| [pandas/io/gbq.py](https://codecov.io/gh/pandas-dev/pandas/pull/26279/diff?src=pr&el=tree#diff-cGFuZGFzL2lvL2dicS5weQ==) | `78.94% <0%> (-10.53%)` | :arrow_down: |
| [pandas/core/frame.py](https://codecov.io/gh/pandas-dev/pandas/pull/26279/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvZnJhbWUucHk=) | `96.9% <0%> (-0.12%)` | :arrow_down: |
| [pandas/util/testing.py](https://codecov.io/gh/pandas-dev/pandas/pull/26279/diff?src=pr&el=tree#diff-cGFuZGFzL3V0aWwvdGVzdGluZy5weQ==) | `90.71% <0%> (+0.1%)` | :arrow_up: |

------

[Continue to review full report at Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/26279?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/26279?src=pr&el=footer). Last update [f46ab96...a402489](https://codecov.io/gh/pandas-dev/pandas/pull/26279?src=pr&el=lastupdated). Read the [comment docs](https://docs.codecov.io/docs/pull-request-comments).
 Thanks @topper-123 !