Firstly, I would like to explain the different ways to achieve a Chakra Mode.
The fact that both Ten-Tails Chakra users and Tenseigan users have Chakra Mode and both are Gen 2 (directly related to Kaguya), could possibly mean that you'll get Chakra Mode if you achieved one of the Gen 2 abilities:
- Ten-Tails Chakra (Kurama, Madara, Obito, and Naruto)
- Rinnegan AND Six Paths Sage Mode (Hagoromo)
- Tenseigan (Hamura and Toneri)
Kurama had that chakra avatar which is basically him going Chakra Mode, Madara had that white cloak Chakra Mode, Obito had those red chakra arms, which I consider that a partial Chakra Mode, Naruto had that "Ten-Tails Chakra Mode", and Toneri had that Tenseigan Chakra Mode.
The above ability is of course a next-step ability based on another speculation I've observed, which is that Gen 3 (directly related to Hagoromo and his descendants) people are split into 2 groups (Indra, Asura), with each group having a Chakra Structure ability (Susanoo, Adamantine Chain). Logically, this would be a watered down ability of the Chakra Mode mentioned above, which completes the evolutionary chain of the chakra-related abilites.
Secondly, I would like to explain the different Tailed Beast Chakra Modes of Naruto.
Naruto had a total of 8 different Chakra Modes throughout the series, and the ones after the war are the msot confusing. But if we separate them into different versions (Manga/Anime/Movie, Gaiden, Boruto), then it would be easy to understand.
- Chakra Mode (Chakra only)
- Link Mode (Chakra and heartbeat)
Minato's Chakra Mode and Naruto's first Chakra Mode are the Yin and Yang half of a complete Chakra Mode that first appeared in The Last. This complete form then reappeared albeit slightly changed in the next movie when Naruto first fought Momoshiki. If you compare the two you'll see the similarities (opened circle on the stomach).
The Kurama Link Mode that Naruto used in the war also reappeared albeit slightly changed in the movie when Naruto fights Momoshiki for the second time, probably due to Naruto having both Yin and Yang Kurama. Again, if you compare the two you'll see the similarities (full circle on the stomach, black undershirt with a curve bottom).
The mode that Naruto used in Gaiden and in Boruto is their respective versions of the mode mentioned in the previous paragraph, hence not having as much similarities as the modes previously mentioned.
The mode that Naruto used to fight Madara, Kaguya and subsequently Sasuke is the Link Mode that utilised all the Tailed Beasts' Chakra, with a signature black undersuit.
Lastly, I would like to explain the relationship between Tailed Beast Chakra Mode and Jinchuriki Forms.
There are 3 types of Jinchuriki Forms, first one is a human using the chakra (Initial, Version 1), second one is a hybrid between a human and a Tailed Beast (Version 2), and the last one is the Tailed Beast (Partial, TB Mode).
There are also 3 types of Tailed Beast Chakra Mode, first one is a human using the chakra (Nine-Tails Chakra Mode), second one is a human linking their heartbeat with a Tailed Beast (Kurama Link Mode), and the last one is the Tailed Beast (Kurama Mode).
Basically, if you see the similarities, the 3 types of Tailed Beast Chakra Mode is just the 3 types of Jinchuriki Forms but with the addition of physically becoming chakra, which could be due to Kurama being the largest part of the Ten-Tails Chakra.