Why doesn't the translation save when I manually translate meta title or meta description? (TMS)

One of the common reasons is the character limitation of meta title and meta description.

Meta title can have up to 70 characters, and meta description can have up to 320 characters. If the translation text is more than the max character limitation, the extra characters will be trimmed automatically. That might give you an impression the translation text is not saved.

You can find more information about this on this page -https://help.shopify.com/en/manual/promoting-marketing/seo/adding-keywords#edit-the-title-and-meta-description-for-a-page