If a sister donates her brother's money, is that considered a binding vow, or does her informing her brother that she made such a vow not obligate her to anything?

If you did not intend a vow (nadhr), but rather intended to convey something incorrect, then there is no vow upon you, because the Prophet (peace be upon him) said: "Deeds are only by intentions." A vow must include intention. Your telling your brother that you made a vow, when you did not, is considered a lie, so you must repent to Allah. Nothing results from self-talk (hadith al-nafs), because the Prophet (peace be upon him) said: "Indeed, Allah has forgiven my Ummah for whatever thoughts cross their minds, as long as they do not act upon them or speak them." Ignore the whispers (waswas).