REGR: revert behaviour of getitem with assigning with a Series
Closes #37427

@jbrockmendel this is reverting part of the clean-up you did in https://github.com/pandas-dev/pandas/pull/33643. I think the behavioural change was unintentional in that PR. We might still want to do it for 1.2, but then with a deprecation first.
@MeeseeksDev backport to 1.1.x 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 1.1.x
$ git pull
```

2. Cherry pick the first parent branch of the this PR on top of the older branch:
```
$ git cherry-pick -m1 5492ed7d4a9fdc310b26d773baebebce4e0e2dde
```

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

```
$ git commit -am 'Backport PR #37502: REGR: revert behaviour of getitem with assigning with a Series'
```

4. Push to a named branch :

```
git push YOURFORK 1.1.x:auto-backport-of-pr-37502-on-1.1.x
```

5. Create a PR against branch 1.1.x, I would have named this PR:

> "Backport PR #37502 on branch 1.1.x"

And apply the correct labels and milestones.

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

If these instruction are inaccurate, feel free to [suggest an improvement](https://github.com/MeeseeksBox/MeeseeksDev).
                