[TASK] Migrate @typo3/form/backend/form-manager to TypeScript

Resolves: #101634
Related: #82577
Releases: main, 12.4
Change-Id: Ia9ad4d39e925f671707ec62cd989bfca6dbf32ab
Reviewed-on: https://review.typo3.org/c/Packages/TYPO3.CMS/+/80467


Tested-by: Benjamin Franzke <ben@bnf.dev>
Reviewed-by: Benni Mack <benni@typo3.org>
Tested-by: Andreas Nedbal <andy@pixelde.su>
Reviewed-by: Andreas Nedbal <andy@pixelde.su>
Tested-by: Andreas Fernandez <a.fernandez@scripting-base.de>
Tested-by: Christian Kuhn <lolli@schwarzbu.ch>
Reviewed-by: Benjamin Franzke <ben@bnf.dev>
Reviewed-by: Oliver Hader <oliver.hader@typo3.org>
Reviewed-by: Andreas Fernandez <a.fernandez@scripting-base.de>
Reviewed-by: Benjamin Kott <benjamin.kott@outlook.com>
Tested-by: core-ci <typo3@b13.com>
Tested-by: Benjamin Kott <benjamin.kott@outlook.com>