Will a person who donates to a charity project out of fear of his direct manager, and not with a sincere intention, be rewarded with the same reward as someone who donates out of his own free will and good intention?

You will not be rewarded for this amount, because you did not intend it for the sake of Allah. Rather, you offered it out of fear of your companion. And because the Prophet, peace and blessings be upon him, said: "Actions are but by intentions, and every man shall have but that which he intended."