Same name of the demon with different rarity is still treated as same demon.
With this, you are not able to put them together in the same party.
You may use "Aether Crystal" to awaken the characters other than using its respective aether
BAYONETTA Collaboration Characters：「Witch Aether S」「Witch Aether L」
DMC5 Collaboration Characters：「Soul Aether S」「Soul Aether L」
Berserk Collaboration Characters：「Od Aether S」「Od Aether L」
GHOST IN THE SHELL: SAC_2045 Collaboration Characters：「Cyber Aether S」「Cyber Aether L」
You may also check the specific details regarding the collaboration characters on the following links
BAYONETTA Collaboration Characters