There are three parts to this answer:
- Sodium borohydride is more basic than alcohols
- Sodium borohydride reacts with anhydrides under some conditions
- Reactions can be autocatalytic
Sodium borohydride is more basic than alcohols
Sodium borohydride will react with water and alcohols to form small amounts of hydrogen gas. If you doubt this, add some to water or ethanol or try to find a video of this phenomenon.
$$\ce{NaBH4 + ROH -> NaOR + BH3 + H2 ^}$$
If sodium borohydride can deprotonate an alcohol, it is more basic than an alcohol.
Sodium borohydride reacts with anhyrides under some conditions
Here is an exmaple In this article in the Journal of the American Chemical Society, sodium borohydride in THF is used to reduce cyclic anhydrides to lactones (esters), not your expected product! However, this reaction might not be general to all anhydrides. Introductory textbooks tend to try to make things simpler to help you learn the material. Think about how maddening it would be to try to learn all of the conditions in which sodium borohydride reacts with anhydrides and all of the ones that do not work.
Reactions can be autocatalytic
If a reaction produces a species as a product that can catalyze the reaction, the rate may increase over time instead of decrease. These reactions are autocatalytic.
Consider the reaction of an alcohol with an anhydride.
$$\ce{ROH + (R'CO)2 -> ROCO2R' + R'CO2H}$$
As you mention, acid is not necessary. However, the substitution reactions of carboxylic acid derivatives are all accelerated by acid.
Rate law without acid $$\mathrm{rate}=k[\ce{ROH}][\ce{(R'CO)2O}]$$
Rate law with acid $$\mathrm{rate}=k[\ce{ROH}][\ce{(R'CO)2O}][\ce{HA}]$$
As the reaction proceeds, a carboxylic acid is produced. While it is not a particularly strong acid, the reaction mixture does become more acidic over time, and the reaction rate increases as a result. This is one reason why alcohols (seemingly poor nucleophiles) react with anhydrides on their own.