As Tadashi and Elsa are from different series, they have never met in canon. Even though a few Disney crossover video games could have the two cross paths with one another or get them added in the same team, by the player's choice.
Both Elsa and Tadashi are oldest in their families, who care a great deal about their young siblings and had lost their parents. Which has the two looking out for Anna and Hiro and even serve as their carers, making it the main reason why they can be a bit over protective in their own ways. Since Elsa had accidently struck Anna's head with her powers and didn't want it to happen again, by giving up her own freedom, while Tadashi doesn't like Hiro taking part in the illegal Bot Fights and does want he could to set Hiro onto the right track. The Big Hero 6 manga even had Tadashi striking Hiro in the face when they had a fight about Tadashi leaving, while the Frozen film has Elsa fighting with Anna about keeping herself away from her and the outside world, along the Frozen film and the BH6 manga having Elsa and Tadashi keeping secrets from Anna and Hiro until they learned what they are without their say so. Even though certain events has Elsa and Tadashi leaving Anna and Hiro's side, a part of them still remains with their siblings that keeps them going.
Elsa has magical powers over ice and had lost control over them until Anna helps Elsa to master them again through love, while Tadashi had created a snow making machine which malfunctioned when he used it for the first time, but after looking through Hiro's note Tadashi was able to make it work and decides to give it to his brother as a Christmas. Just as Big Hero 6 has a Christmas episode, Frozen has its own short Christmas film which who feature the bond and love that Elsa and Tadashi have for Anna and Hiro. Tadashi's science and Elsa's magic is what allowed them to create and live life to Baymax and Olaf, who both have a large white and a very small black color scheme and like giving others hugs that can be warm. Tadashi and Elsa have also used their abilities and skills to bring joy to others and their family, even when they and their siblings sometimes get into trouble with their parents/guardian. Tadashi's aunt has a pet cat, while one of Elsa's friends with very close with his reindeer. Because Elsa and Tadashi are the eldest siblings, they remember their parents during their child years a lot more better than Anna and Hiro do.
While the main antagonists in their Disney films, Hans and Yokai as Callaghan, had presented themselves as a friend to Elsa and Tadashi while hiding their dark sides from them and their friends. Since Callaghan had left Tadashi to die when he tried to save the man, while before Hans had tried to kill Elsa he had saved her live from the Duke's men. Hans and Callaghan showing their ugly sides to their younger siblings had shocked when, along with the two villains showing no remorse for their actions towards Elsa and Tadashi. There have even been times where their home city/kingdom, San Fransokyo and Arendelle, were nearly destroyed by an unnatural force.
Tadelsa is considered to be the most well liked het ship of the Frozen/Big Hero 6 crossover fandom. Along with it being one of the many ships to be rivaled with Jelsa and possibly Jadashi. Since both of their Disney films are commonly crossed with Rise of the Guardians, together or on separate occasions, where Tadashi is commonly featured as a Guardian with fire related abilities while Elsa is sometimes featured as a second ice wielding Guardian of Childhood. The fanon idea of Tadashi having his own elemental power, even if it is the opposite to ice, is sometimes used in Tadelsa. There have even been times where both Elsa and Tadashi have been added to the Rise of the Brave Tangled Dragons crossover fandom, after it was renamed to Rise of the Brave Tangled Frozen Dragons.
Tadashi's kind nature and desirer to help others, would have him not being scared of Elsa or her powers, he learns that she is good person who doesn't want to hurt anyone, he would want to help her. Since Tadashi is a scientist he would probably be more fascinated than scared of Elsa's abilities and may want to learn more about them, with the promise to help Elsa gain a better understanding of them in hopes they'll able to find away which allow Elsa to regain control over them again. Fans even seen them bonding over the struggles of being the eldest in their families when it comes to their younger siblings, along with sharing stories of the fun times they had shared with Hiro and Anna.
Even though there are times where the two are featured in an Arendelle setting and timeline, Tadelsa, and the other Frozen/Big Hero 6 crossover ships, commonly has them in a modern setting where both Tadashi and Elsa reside in San Fransokyo. Which might be why the two have also been featured in an Asian setting, since Tadashi is half Japanese, there being a few Japanese themes used in the Big Hero 6 franchise and there have been times where fans place Elsa in a Kimono or Asian style clothing and setting in their fan art. A few manips of the ship's modern setting sometimes has Elsa in Go Go's clothing, which might be a nod to Tomadashi and possibly Elsogo, or when fans have Elsa in Go Go's dark clothes or a blue version of her armour suit in their fan manip work on the Frozen Hero 6 crossover fandom.
Because of BH6's superhero theme and modern timeline setting, Tadashi and Elsa are commonly featured in a Superhero AU, that is commonly of the Big Hero 6 universe and have Tadashi has a member of the team, while Elsa is viewed as a civilian girl who Tadashi entrusts his superhero secret with or keeps her out of it in order to protect Elsa. In a few versions of a Frozen AU, with the BH6 characters, Tadashi is sometimes seen as male Elsa, while in Big Hero 6 AU with characters Elsa is sometimes seen as a female Tadashi Hamada, whenever they and their younger siblings are viewed as male versions of the Arendelle Sisiters and females versions of the Hamada Brothers. As for a few versions of a Dark AU, Elsa is commonly viewed as her antagonist self from one of the earlier layouts of Frozen, while Tadashi's dark alter ego is based on the fan-theory of him being Yokai and possibly the dark out comes to what might had happened should Tadashi had survive the fire.
On AO3, the ship only has 14 fanfics while fanfiction.net has 42 with only 17 placed under pairing. Its fanbase on Deviantart, however, it a bit more larger than it is on the two fanfic websites.
- Elsa makes a appearance in Kingdom Hearts III, while Tadashi was only mentioned in it.
- A few Frozen Easter eggs have been seen in the Big Hero 6 film.
- In "The Present", a reference to the Frozen song, "Do You Want to Build a Snowman?", was made when Honey Lemon asks the villain if they want to build a snowman.