For Gen 9's third legendary

Looking the legendary duos, trios and quartets:

1. Gen I: Ice/Electric/Fire, Psychic
2. Gen II: Fire/Electric/Water, Fire/Psychic
3. Gen III: Rock/Ice/Steel, Ground/Water/Flying
4. Gen IV: Dark/Psychic, Steel/Water/Ghost, 
5. Gen V: Flying/Electric/Ground, Rock/Grass/Steel/Water, Electric/Fire/Ice
6. Gen VI: Dark/Fairy/Ground
7. Gen VII: Electric/Grass/Psychic/Water, Steel/Ghost/Dragon
8. Gen VIII: Fairy/Fighting/Poison

Using the current corresponding TCG types:

1. Gen I: Water/Electric/Fire, Psychic
2. Gen II: Fire/Electric/Water, Fire/Psychic
3. Gen III: Fighting/Water/Steel, Fighting/Water/Normal
4. Gen IV: Dark/Psychic, Psychic, Steel/Water/Psychic, 
5. Gen V: Normal/Electric/Fighting, Fighting/Grass/Steel/Water, Electric/Fire/Water
6. Gen VI: Dark/Fairy/Fighting
7. Gen VII: Electric/Grass/Psychic/Water, Steel/Psychic/Dragon
8. Gen VIII: Fairy/Fighting/Dark

Shared types:

1. Gen I: Flying
2. Gen II: Flying
3. Gen III: N/A (Fire*/None/Dragon)
4. Gen IV: Psychic, Dragon
5. Gen V: Flying, Fighting, Dragon
6. Gen VI: N/A(Flying/None/Dragon)
7. Gen VII: Fairy, Psychic
8. Gen VIII: N/A?(Steel**/Steel**/Dragon)

*Primal Reversion only.

**Alternate form only.

Looking at this chart, it seems next generation's third legendary will be Fairy/Dragon or will belong to one of the elemental types besides Water. We are also ignoring the possibility of a mono Dragon outside of another type change gimmick.

**It cannot be Rock or Fighting since Zygarde covers both in TCG.

**It cannot be Dark since Eternatus covers it in TCG as Poison is changed to Darkness in TCG this gen.

**It cannot be Normal since Rayquaza covers it in TCG.

**It cannot be Water since Kyurem covers it in TCG.

**It cannot be Psychic, since both Giratina and Ultra Necrozma cover it in TCG.

**It can be Electric, Fire or Steel, but those are done for non-third legendary mascot legendaries.

**It can be Grass, I just don't expect it because of obvious reasons(even though Grass/Dragon would make sense in a Chinese region).

What happens after that is ambiguous. Since next generation will likely bring one of the existing types to TCG like what happened to Dragon in Gen V, and the next third legendary in Gen 10 will depend on that. If it is Ghost or Bug, then we will likely move onto to regular Dark and Fighting, as those are covered by existing Dragon pseudos(similar to what happened to Zygarde). If it is Flying, Ice or Rock/Ground, then the remainder type (Normal/Water/Fighting) will likely be the candidate.

Edit (10.10.2020):

https://www.pokebeach.com/2020/01/nine-surprising-changes-announced-for-the-ptcg-fairy-pokemon-dropped
Fairy-type is now integrated into Psychic-type and ceases to be its own thing.
Edit (12.01.2020): Since Fairy-type is now integrated into Psychic-type, the updated chart looks like this:
Using the current corresponding TCG types:

1. Gen I: Water/Electric/Fire, Psychic
2. Gen II: Fire/Electric/Water, Fire/Psychic
3. Gen III: Fighting/Water/Steel, Fighting/Water/Normal
4. Gen IV: Dark/Psychic, Psychic, Steel/Water/Psychic,
5. Gen V: Normal/Electric/Fighting, Fighting/Grass/Steel/Water, Electric/Fire/Water
6. Gen VI: Dark/Psychic/Fighting
7. Gen VII: Electric/Grass/Psychic/Water, Steel/Psychic/Dragon
8. Gen VIII: Psychic/Fighting/Dark

Looking at this new chart, we can say that Fairy/Dragon is not happening. Instead, Grass/Dragon, Steel/Dragon or Electric/Dragon is now likely. If it is Grass, then the rest will likely be Electric and Water since we have Fire and Water(ORAS) and Electric and Fire(BW). If it is Steel, then the rest will be according to Regis' types. If it is Electric, rest can be Flying and Ground, similar to Kamis.