In Mexico we usually use these words when we want to put appart chicken, pork or any other type of meat. But I´m not sure if both can be used in the same way
The words mean slightly different things.
Desmenuzar means "to break something apart by dividing it into small parts." Deshebrar means "to break something into very thin parts, like fibers." (Both translated from the linked DLE entries.) Desmenuzar refers to menudo "minute, very small", so you can use with a clump of dirt or a piece of stale bread, for example, that crumbles when you apply force to them between your fingers. Deshebrar directly refers to hebras "fibers", so it specifically means "to tear apart in thin strips", which you can only do with something that has fibers and that you can cut lengthwise (like meat).
I never use deshebrar (I'm from Argentina) but it seems to me that desmenuzar is more general. This is the verb we sometimes use with chicken, in particular, though mostly we prefer trozar.
As, pablodf76 points out, yes, they have different meanings. But I think that context can be important when comparing this words. Let's look at the definitions in the Diccionario del Español de México:
v tr (Se conjuga como amar) Fragmentar o desintegrar en pedazos pequeños, generalmente con las manos y sin utilizar ningún instrumento cortante. “Desmenuce el bacalao”, “Después de desmenuzarlo, eche el pollo al caldo”, “El arado iba desmenuzando los terrones”.
v tr (Se conjuga como amar) Sacar las hebras de una tela o hacer hebras de un tejido denso, como la carne.
For example, you can say
Desmenucé el pan, but it doesn't make sense saying
Deshebré el pan. However, in our context (Mexico), both words can be perfectly interchangeable when talking about shredding meat.
Desmenucé el pollo and
Deshebré el pollo (in México) mean the same thing: people will understand that you shredded/pulled apart the meat.