The terminology used by the internal image IPTC data is Keywords, not tags. The latter are what the Finder assigns as names representing organizational colors that appear next to the filename in the Finder.
The Finder offers to show you EXIF data from the image, but does not offer IPTC information in any view mode, even in Preview mode.
Spotlight will match images that contain a specific keyword, but before that can happen, the drive needs to be indexed by Spotlight on its Image search results category. Your external drive may not be compatible with Spotlight if it is not FAT32, HFS+, or APFS formatted.
Once indexed, an image whose IPTC keyword was set in Preview to "Birthday pix" can be found with Spotlight as:
kind:image +.jpg keyword:birthday
kind:image +.jpg keyword:"birthday pix"
Even when indexed, the command-line mdls(1) tool will not show the IPTC keyword(s).
If the drive cannot be indexed by Spotlight, then you will need to add the image to Photos where the keyword will appear below the Add a Description section, when you select Get Info on the image. Otherwise, use a free, third-party tool (e.g. ExifTool) to get at the IPTC data.