To see the file name of an image or video in the Apple Photos app on iOS, open the app, tap on the media, then swipe up or tap the info icon to reveal the assigned file name.
When sharing media and saving it to Files, the file name will be displayed at the bottom of the Files picker.
Media saved in Photos follows the default naming scheme 'IMG_X' where X is a counter, except for media from AirDrop, Mail, Messages, or third-party apps.
To change the file name, save the media to Files, then enter a new name or export it to rename it later.
Photos doesn't support searching by file names but allows searches by keywords, captions, or complex descriptions.