The terms get used loosely, but they describe two different products with two different jobs.
What a mixer is
A mixer is one component of a cocktail. Tonic water, sour mix, a bloody mary mix. You still need to combine it with spirit, and often with other mixers or garnishes, to get a finished drink. The recipe work is still on you.
What a batch cocktail is
A batch cocktail is the finished drink profile, already balanced across every ingredient the recipe calls for, made in volume. Pour it over ice, add spirit if desired, garnish, done. The recipe work is already finished.
Why the difference matters for cost and waste
Mixers often come in large shelf-stable bottles designed for single-ingredient use across many different drinks, which means less waste per bottle but more labor per drink. Batch cocktails are built for one specific drink, made fresh, with a shorter shelf life but zero labor to build the final pour.