Monoma and Shinso are quite similar, as both have quirks that society as a whole has deemed "unheroic," and instead of aiding them; their quirks been a cause of their troubles. In Monoma's case, he sees himself and his classmates as superior to Class 1-A, however, he has remained popular among his class due to his smarts and charisma.
They meet for the first time in the Joint Training Arc and Monoma immediately takes a liking to Shinso due to their shared disdain for Class 1-A. Throughout the arc, he says things like "I like this one" (referring to Shinso) and "I've really taken an interest in you," to Shinso. During training, they are sorted into the same team, Team 5-B.
When it is their team's turn to battle against Team 5-A, Shinso suggests combining his and Monoma's quirks to defeat the opposing team, to which Monoma responds that it worries him to be relied on and that he will be looking to Shinso.
Monoma brings up their similarities during the battle, telling Shinso that since their quirks don't have any raw (physical) power, they need to "embrace their darker side," in order to be useful in battle.
When Midoriya's Blackwhip goes berserk, Monoma immediately calls out to Shinso in worry. And when Monoma is caught by Uraraka, Shinso calls out to him while thinking, "I need to save him."
It's shown that the two do care for one another, as they both have apologetic inner monologue, disappointed that they weren't help each other out more. Monoma thinks an apology to Shinso about getting himself caught and Shinso apologizes in his mind to Monoma for retreating instead of saving him.
Monoma helps out Shinso by using Nirengeki Shoda's Twin Impact Quirk on Midoriya before Midoriya could attack Shinso. To this, Shinso praises Monoma, thinking "Well done, Monoma!"
Monoma and Shinso don't have too many appearances in merch or in various games, but when they do, they are often together.
Before Monoma and Shinso had even canonically interacted in the story, fans began shipping them together for their mutual dissent towards Class 1-A and their "villainous"-style Quirks. In time, the ship started gaining more attention following the release of the Joint Training Arc chapters.
Shippers often portray Monoma as the one who has fallen for and begins to pursue Shinso, while Shinso apathetically accepts and eventually appreciates Monoma's over-the-top behavior.