How do you know that a given reaction is not an acid base reaction? (apart from the fact that it should have an acid and a base and it should be a double displacement reaction)?

I mean I know acids have protons (H+) and bases have hydroxides (OH-) but there are so many other acids and bases that don’t have H+ and OH-.

I think your problem may be that you simply must recognize that there are SEVERAL definitions.

An acid/base pair by the Arrhenius definition may not be an acid/base pair by the Lewis definition which may not be an acid/base pair by the Bronsted-Lowry definition, which may .........and so on into the night. Does this answer your question or have I taken your remarks too literally.
By the way, when I was in school we were taught that all acid base reactions were double displacement and produced a salt + water BUT what about
HCl + NH3 ==> NH4Cl
HCl is an acid and NH3 is a base (both by the Bronsted-Lowery definition) and that's a synthesis reaction.

To determine if a given reaction is not an acid-base reaction, there are certain criteria you can consider apart from just looking for an acid and a base or a double displacement reaction. While it is true that not all acids and bases contain H+ and OH-, there are other characteristics that can help you identify if a reaction is not an acid-base reaction. Here are a few factors to consider:

1. Reactive Species: In an acid-base reaction, the active species involved are usually proton donors (acids) and proton acceptors (bases). Look for molecules or ions that can donate or accept protons. Examples include acid-base conjugate pairs like HCl/Cl- or NH4+/NH3.

2. pH Change: Acid-base reactions typically involve the transfer of protons, which leads to a change in the pH of the solution. If the reaction you are examining does not result in a pH change, it is likely not an acid-base reaction.

3. Conjugate Acid-Base Pairs: Acid-base reactions involve the formation and dissociation of conjugate acid-base pairs. Look for species that have the ability to form conjugate pairs by gaining or losing protons. For example, ammonia (NH3) can act as a base by accepting a proton and forming ammonium ion (NH4+).

4. Proton Transfer: Acid-base reactions involve the transfer of protons from one species to another. If the reaction you are considering does not involve any proton transfer, it is not likely an acid-base reaction.

Remember that these criteria are not exhaustive, and the presence of a specific acid or base does not guarantee that a reaction is an acid-base reaction. It is important to analyze the reaction as a whole, considering the changes in species, pH, and proton transfer to determine if it fits the characteristics of an acid-base reaction.