Kaeya and Dainsleif are yet to interact in the canon and as far as we know, they do not know each other. They both have ties to the ancient nation of Khaenri'ah. Dainsleif was a royal guard and Kaeya is implied to be a descendent from there. This could mean that Dainsleif knows of or was once acquainted with Kaeya's ancestry, or perhaps even Kaeya himself. This seems to be the main reason people theorize they are connected.
During "Bough Keeper: Dainsleif"—the quest that features Dainsleif's first in-game appearance— Katheryne tells the traveler about a mysterious stranger (Dain). She describes him, saying he "is a very tall man, and even has an eyepatch". Upon hearing this, the traveler says "So, just some distant relative of Kaeya's, then". Katheryne replies, "That stranger declined the Guild's invitation, so I can't say that I know anything about the relationship between him and Mr. Kaeya." Although the quest dialogue mentions Kaeya briefly, this can be interpreted as foreshadowing, and alludes to the fact Kaeya and Dainsleif have a similar background and/or might have some sort of relation to each other.
Dainsleif and Kaeya have some similarities in appearance, namely the abyss stars in their eyes. They also have hues of blue in their color palettes (along with black, grey, and white). Personality-wise, they are fairly different. Dainsleif is more stoic and reserved, whereas Kaeya is more outgoing and charming. These characters are both enigmas in the eyes of the traveler however, and have yet to reveal certain information about themselves/their past in game.
DainKae is a rarepair within the Genshin community because of their lack of clear connection. Many who ship it believe in the Kaeya prince theory in which Kaeya is speculated to be the prince of Khaenri'ah. This would give the two a clear connection as Dainsleif was canonically a royal guard for the nation. This troupe of a prince and someone of lower status falling in love is typically the idea of the ship though many simply think their yin and yang personalities would be interesting together. Many theorize that Dainsleif would be the one to call Kaeya back to Khaenri'ah meaning they would reunite.
There is a sizable amount of fan work on them ranging from fanfictions to art. However, Mihoyo has yet to produce art of the two together. DainKae currently has 151+ works on Archive of our own. This ship has some rivals despite the popularity, which are KaeLumi, Kaether, KaeLuc, KaeLisa, HalfDain, Kaebedo and DainsLumi.
- Dainsleif and Kaeya have a similar color palette
- Because of their relations to Khaenri'ah, Dainsleif and Kaeya both have the Abyss stars in their eyes, and have coverings on their eyes (Kaeya with eyepatch; Dain with his eye mask)