REF: Simplify Datetimelike constructor dispatching
Implement several missing tests, particularly for TimedeltaArray

Move several things to DatetimeLikeArrayMixin that will need to be there eventually.

Misc cleanups.
Hello @jbrockmendel! Thanks for submitting the PR.

 - There are no PEP8 issues in the file [`pandas/core/arrays/datetimelike.py`](https://github.com/jbrockmendel/pandas/blob/eae838968abaadbade999d437f07f4876d356468/pandas/core/arrays/datetimelike.py) !

 - There are no PEP8 issues in the file [`pandas/core/arrays/period.py`](https://github.com/jbrockmendel/pandas/blob/eae838968abaadbade999d437f07f4876d356468/pandas/core/arrays/period.py) !

 - There are no PEP8 issues in the file [`pandas/core/arrays/timedeltas.py`](https://github.com/jbrockmendel/pandas/blob/eae838968abaadbade999d437f07f4876d356468/pandas/core/arrays/timedeltas.py) !

 - There are no PEP8 issues in the file [`pandas/core/indexes/datetimelike.py`](https://github.com/jbrockmendel/pandas/blob/eae838968abaadbade999d437f07f4876d356468/pandas/core/indexes/datetimelike.py) !

 - There are no PEP8 issues in the file [`pandas/core/indexes/datetimes.py`](https://github.com/jbrockmendel/pandas/blob/eae838968abaadbade999d437f07f4876d356468/pandas/core/indexes/datetimes.py) !

 - There are no PEP8 issues in the file [`pandas/core/indexes/period.py`](https://github.com/jbrockmendel/pandas/blob/eae838968abaadbade999d437f07f4876d356468/pandas/core/indexes/period.py) !

 - There are no PEP8 issues in the file [`pandas/core/indexes/timedeltas.py`](https://github.com/jbrockmendel/pandas/blob/eae838968abaadbade999d437f07f4876d356468/pandas/core/indexes/timedeltas.py) !

 - There are no PEP8 issues in the file [`pandas/io/pytables.py`](https://github.com/jbrockmendel/pandas/blob/eae838968abaadbade999d437f07f4876d356468/pandas/io/pytables.py) !

 - There are no PEP8 issues in the file [`pandas/tests/arrays/test_datetimelike.py`](https://github.com/jbrockmendel/pandas/blob/eae838968abaadbade999d437f07f4876d356468/pandas/tests/arrays/test_datetimelike.py) !

 - There are no PEP8 issues in the file [`pandas/tests/indexes/datetimelike.py`](https://github.com/jbrockmendel/pandas/blob/eae838968abaadbade999d437f07f4876d356468/pandas/tests/indexes/datetimelike.py) !

 # [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/23140?src=pr&el=h1) Report
> Merging [#23140](https://codecov.io/gh/pandas-dev/pandas/pull/23140?src=pr&el=desc) into [master](https://codecov.io/gh/pandas-dev/pandas/commit/913f71f74709e4d4814e50f21768b97ce2c6ef8d?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/23140/graphs/tree.svg?width=650&token=eZ4WkYLtcO&height=150&src=pr)](https://codecov.io/gh/pandas-dev/pandas/pull/23140?src=pr&el=tree)

```diff
@@            Coverage Diff             @@
##           master   #23140      +/-   ##
==========================================
+ Coverage   92.19%   92.19%   +<.01%     
==========================================
  Files         169      169              
  Lines       50959    50986      +27     
==========================================
+ Hits        46980    47009      +29     
+ Misses       3979     3977       -2
```

| Flag | Coverage Δ | |
|---|---|---|
| #multiple | `90.62% <100%> (ø)` | :arrow_up: |
| #single | `42.28% <49.18%> (-0.02%)` | :arrow_down: |

| [Impacted Files](https://codecov.io/gh/pandas-dev/pandas/pull/23140?src=pr&el=tree) | Coverage Δ | |
|---|---|---|
| [pandas/core/indexes/period.py](https://codecov.io/gh/pandas-dev/pandas/pull/23140/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvaW5kZXhlcy9wZXJpb2QucHk=) | `93.22% <ø> (-0.02%)` | :arrow_down: |
| [pandas/core/arrays/datetimelike.py](https://codecov.io/gh/pandas-dev/pandas/pull/23140/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvYXJyYXlzL2RhdGV0aW1lbGlrZS5weQ==) | `94.97% <100%> (+0.07%)` | :arrow_up: |
| [pandas/core/indexes/datetimelike.py](https://codecov.io/gh/pandas-dev/pandas/pull/23140/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvaW5kZXhlcy9kYXRldGltZWxpa2UucHk=) | `98.25% <100%> (+0.02%)` | :arrow_up: |
| [pandas/core/arrays/timedeltas.py](https://codecov.io/gh/pandas-dev/pandas/pull/23140/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvYXJyYXlzL3RpbWVkZWx0YXMucHk=) | `94.47% <100%> (+0.5%)` | :arrow_up: |
| [pandas/compat/numpy/function.py](https://codecov.io/gh/pandas-dev/pandas/pull/23140/diff?src=pr&el=tree#diff-cGFuZGFzL2NvbXBhdC9udW1weS9mdW5jdGlvbi5weQ==) | `87.97% <100%> (+1.31%)` | :arrow_up: |
| [pandas/core/indexes/timedeltas.py](https://codecov.io/gh/pandas-dev/pandas/pull/23140/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvaW5kZXhlcy90aW1lZGVsdGFzLnB5) | `90.65% <100%> (-0.12%)` | :arrow_down: |
| [pandas/core/arrays/period.py](https://codecov.io/gh/pandas-dev/pandas/pull/23140/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvYXJyYXlzL3BlcmlvZC5weQ==) | `95.97% <100%> (+0.4%)` | :arrow_up: |
| [pandas/io/pytables.py](https://codecov.io/gh/pandas-dev/pandas/pull/23140/diff?src=pr&el=tree#diff-cGFuZGFzL2lvL3B5dGFibGVzLnB5) | `92.44% <100%> (ø)` | :arrow_up: |
| [pandas/core/indexes/datetimes.py](https://codecov.io/gh/pandas-dev/pandas/pull/23140/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvaW5kZXhlcy9kYXRldGltZXMucHk=) | `96.47% <100%> (-0.03%)` | :arrow_down: |

------

[Continue to review full report at Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/23140?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/23140?src=pr&el=footer). Last update [913f71f...b5827c7](https://codecov.io/gh/pandas-dev/pandas/pull/23140?src=pr&el=lastupdated). Read the [comment docs](https://docs.codecov.io/docs/pull-request-comments).
 > can u just make the validation for axis generic

See joris's comment above. The non-controversial parts of this have been ported to separate PRs.  Closing.