It would depend on the situation, if someone is screaming black lives matters in anger and you respond with all lives matter, both statements are used as a weapon. If one person makes the statement that all lives matters it covers black lives too. Taken by itself, black lives matters statement does not have to mean that other lives do not. Words and situation matters too.
I see anyone responding to the black lives matter with a smile and following up with all lives matter as an agreement with an addition.
I would see an angry response to all lives matter as a sign of intolerance and the angry person as a fool. The logic fallacy would fall on the angry person for giving themselves the power to read the other person mind and worse acting on it.