BUG: Do not use string Index like Datetimelike Index
Follow up of conversation in https://github.com/pandas-dev/pandas/pull/32739#issuecomment-602271577.

The goal is to prevent index of string that look like datetimelike to be used as datetimelike.

- [ ] closes #xxxx
- [x] tests added / passed
- [x] passes `black pandas`
- [x] passes `git diff upstream/master -u -- "*.py" | flake8 --diff`
- [ ] whatsnew entry

so the point is that you _dont_ want `pd.Index(["2020-04-15"])` to be a `DatetimeIndex`?  That's a breaking change.

cc @jorisvandenbossche @TomAugspurger xref #33558 I'd want to deprecate that behavior. `pd.Index(["2020-04-15"])` is already not a `DatetimeIndex`, but, when compared with one, it is cast as one.

If we are to deprecated this behavior, I have some question on how the depreciation it is expected to behave.
Should it show a warning when the Index is cast?
To retains the old behavior, it suffice to cast the Index to DatetimeIndex (i.e with pd.to_datetime), but should we converse the old behavior with a warning that it won't behave the same in the future, or switch to the new behavior directly, with a warning on how to re achieve  the old behavior? So to make it explicit, currently we have this behaviour:

```
In [1]: idx = pd.Index(["2020-04-15"]) 

In [2]: idx     
Out[2]: Index(['2020-04-15'], dtype='object')

In [3]: dtidx = pd.DatetimeIndex(["2020-04-15"])      

In [4]: dtidx 
Out[4]: DatetimeIndex(['2020-04-15'], dtype='datetime64[ns]', freq=None)
```

So a object dtype index with strings, and a DatetimeIndex. 

Comparing them elementwise correctly gives False values:

```
In [6]: idx == dtidx   
Out[6]: array([False])
```

but calling the `equals` method actually returns True:

```
In [5]: idx.equals(dtidx)  
Out[5]: True
```

And I think we want this to give False. Correct?

The same also happens for other "coercible" values (not just strings), at least when DatetimeIndex is the calling index. For example:

```
In [7]: idx = pd.Int64Index(dtidx.asi8)    

In [8]: idx.equals(dtidx)    
Out[8]: False

In [9]: dtidx.equals(idx)   
Out[9]: True
```

I would say we also want to deprecate this behaviour? 
If the Index classes are different, the Index objects can't be equal.

So some questions around this:

- Do we already agree we don't want this behaviour of trying to coerce the other Index? I am in favor of not doing coercing, but so not only for strings but eg also integers. However, where to draw the line? (eg what with an object dtype index of Timestamp objects?)
- Do we need to deprecate this, or can we just fix it as a bug fix? Since it seems this was done quite on purpose in the code, I would say we need to deprecate this?
- The above examples are for the `equals` method. But has this further reaching implications? Does this affect the behaviour of other methods? (eg in the linked PR, I saw mention of union/intersection etc methods) we should deprecate this >    Do we already agree we don't want this behaviour of trying to coerce the other Index? I am in favor of not doing coercing, but so not only for strings but eg also integers. However, where to draw the line? (eg what with an object dtype index of Timestamp objects?)

From here  https://github.com/pandas-dev/pandas/blob/b630cdbc33c6e766cc1bfec857b44b04e9db08f8/pandas/core/indexes/datetimelike.py#L882
you can see that many types where already excluded from being coerced.
```
           "floating",
            "mixed-integer",
            "integer",
            "integer-na",
            "mixed-integer-float",
            "mixed",
```
Maybe instead of an exclusion list, we could have an list of which type can could be coerced.

>    The above examples are for the equals method. But has this further reaching implications? Does this affect the behaviour of other methods? (eg in the linked PR, I saw mention of union/intersection etc methods)

Union and intersection behavior follow equals behavior. The same is true for join and merge.
But changed are needed in `pd.Categorical`
```
date = "2020-03-16"
values = [date, pd.to_datetime(date)]
pd.Categorical(values, ordered=False)
# Categories (1, datetime64[ns]): [2020-03-16]
```

 @nrebena i think the idea of this is fine, can you merge master and can re-evaluate @jreback Rebased.
I am not sure how to both deprecate and test the new behavior.
Deprecating could be in https://github.com/pandas-dev/pandas/blob/b630cdbc33c6e766cc1bfec857b44b04e9db08f8/pandas/core/indexes/datetimelike.py#L882 by warning when using `string` typed index eventually. By trying to parametrize, I did see more case of index of different type that are equals.

```python
import pandas as pd 
i1 = pd.Index([1586736000.0]) 
i2 = pd.Index(pd.to_datetime([1586736000.0])) 
print(i1) 
print(i2) 
print(i1.equals(i2))                                                                                              
# Float64Index([1586736000.0], dtype='float64')
# DatetimeIndex(['1970-01-01 00:00:01.586736'], dtype='datetime64[ns]', freq=None)
# True
```
I am not sure this is not the expected behavior either?

 can u merge master and can look again @nrebena can you add a release note Closing as stale, @nrebena let us know if you'd like to reopen