Generate images assets with canvas size

Are you using the term "generate assets" as a general term, or are you specifically using the feature for the open file under File > Generate > Image Assets and naming your layers appropriately to automatically create Image Assets?

If the latter, you will need to add a white/reveal layer mask to each layer that is the size of the entire canvas, or of a consistent size and position for all images. Generator uses the layer mask to set the canvas size of the exported Image Assets.

P.S. There are scripts that can add a reveal all layer mask to every layer.

Assets with many common items
Have you tried using layer comps, then using file >> export >> layer comps to files.  

Assets that are entirely unique
We need to see  your settings, why would you post without showing them??  We have no idea if you are using file >> export as or some other command. Are you new to photoshop?