Does the file in Photoshop show more than one Background Layer in the Layers Panel (Window > Layers)?
Flattening (Layer > Flatten Image) reduces the Layers to one Background Layer.
Some file formats do not support layers to begin with so when saving a copy as a jpg for example it will get flattened.
Should I always 'flatten' an image before saving then in order to make the file smaller?
That would be imprudent in my opinion.
I think a file’s size is far from its most important property as there are also image quality, future editability, …
That said naturally I know one has to make decisions under economic considerations sometimes, but those should probably rather be made on a case-to-case basis.