BUG: unpickling modifies Block.ndim
- [x] closes #37631
- [x] tests added / passed
- [x] passes `black pandas`
- [x] passes `git diff upstream/master -u -- "*.py" | flake8 --diff`
- [x] whatsnew entry

@meeseeksdev backport 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 dbee8faecf9a3ea65a2861e32c12695eae81f3e6
```

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

```
$ git commit -am 'Backport PR #37657: BUG: unpickling modifies Block.ndim'
```

4. Push to a named branch :

```
git push YOURFORK 1.1.x:auto-backport-of-pr-37657-on-1.1.x
```

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

> "Backport PR #37657 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).
                