I can hear that artifact sound (tone) of doubling the voice using TC-Helicon, the older rackmount model TC-Helicon Voice Works called it 'Thicken' and they have the new pedal version called
VoiceTone Double. Both units basically use 'Unison Mode with Enhancement (additional effects option)'. There is also an echo effect to help give it an overall touch of spread/filler.
Antares Duo Evo is another vocal processor that doubles (thicken) the vocals.
Here, go to page and select Audio/Video > Audio and listen to the two vocal demos (1) Female Vocal Dry. (2) Female Vocal - Processed with Duo Evo.