Before the July 16, 2020 release, all messaging images were added via the ‘Image’ content block. After the release, all the old images from the BEE gallery are transferred to the Media manager repository and are added via the Media content block .
So, if an image in a template was added before the July 16 release, you will not see the 'Browse' button to change/replace that image. However, a workaround to resolve this is to delete the image and drop a 'Media' content block in the place and add the image again. Once you'll add the image via Media block, you will be able to see the Browse button.