Batch process results looks different compared to manually load image and apply the same action

Grayscale is tricky. You need to watch your grayscale profiles. Grayscale is color managed in Photoshop just like RGB files. But outside Photoshop, all bets are off - grayscale color management is virtually non-existent.

To get reasonably predictable results outside Photoshop, you need to use a grayscale profile that matches the destination as closely as possible. For screen work, that means Gray Gamma 2.2 or sGray. For print there are many special considerations I won't go into here - but generally, Gray Gamma 2.2 should be a fairly safe choice.

If you have grayscale files without an embedded profile, your working gray gets assigned. That's already a problem, because the Photoshop default dot gain profiles are virtually useless for any practical purpose. They are basically outdated generic profiles for offset print (for which there are much better solutions today).

My general advice is to avoid grayscale if possible. But if you have to, convert to Gray Gamma 2.2 for general use. Make sure that profile is embedded in the file throughout (which may be where your batch trips up).

Could you please post screenshots with the pertinent Panels (Toolbar, Layers, Options Bar, …) visible and the fully expanded Action in the Actions Panel and the Batch settings?