Why can’t I share my recordings through text message or MMS?

Text message and MMS do not show up in the list of supported apps for MP3 because they do not support MP3 files.

SMS is plain text (and only very short plain text)
MMS does support audio such as MIDI file, but not MP3 clips.

However, Hi-Q MP3 Voice Recorder let you send your clips by using other 3rd party apps, like Skype, WhatsApp, SoundCloud, or online storage such as Dropbox, Evernote or Google Drive. It depends on those apps being installed on your phone.