As Edward and Rin are from two different manga/anime series, they have never met in canon.
Both Ed and Rin are older brothers that can be a bit hot-headed, due to their short tempers in which sometimes hides the kindness in their hearts. As whenever either of them see people in danger or are in need of help, despite the ricks that they put themselves in while doing so, they do whatever they can to help those in need. Their short tempers have even gotten their younger, more sensible brothers to act as the older sibling of the two when needed, as well as making sure that their anger states don't get anymore hurt from them. The two have also gotten their bodies altered from making a choice that changes their lives, along with throwing them into the dangerous and darker sides of their worlds. Ed and Rin's other common link that both of them have lost their mothers at a very young age and want nothing to do with their fathers, as Ed's father left him and his brother at a young age and had felt they alone in the world after their mother dies, while Rin's biological father is the demon god Satan and the main reason why so many people want him dead. On the very day Rin had met Satan for the first, he had took control of his foster father's body, who kills himself to prevent Stan from taking Rin away, making it one of the main reasons why Rin seeks to destroy Satan. Rin and his younger twin brother were raised in a monastery, where their foster father was the head priest, while Ed and his younger brother visited a church to see if its Father really does have the Philosopher Stone. Along Ed learning his father is connected by blood to a man known as Father, who is the main villain of his manga/anime series.
As a State Alchemist, Edward is a dog of the military and one of the state's human weapons, in times of war. Rin has also been called a living weapon, as Mephisto Pheles stated this to the Vatican in order to prevent Rin from being executed before he gets the chance to defeat the head of the Illuminati. A few of the demons that Rin fought have took the form of animals, while Ed has battled a few chimeras, animal hybrids that were created through alchemy. As well as both of them taking on immortal beings, that look human on the outside and share blood with them. Since the eight Demon Kings are Rin's half-brothers from his father's side, while the seven sin Homunculus were created by the very Homunculus that was born from Van Hohenheim's blood. In which makes Ed and Al the human brothers of the Homunculus. The title "Fullmetal Alchemist" was given to Edward by the State Military, while Mephisto Pheles had called Rin the "Blue Exorcist" after he used his blue flames to destroy the Impure King. Rin's signer weapon is the sword that holds his demon heart, while Ed's is the blade he commonly transmute on his auto-mail arm. Even though Ed's singer ability is to transmute alchemy without the need for a transmutation circle, while Rin's is the blue color flames he inherited from his biological father. Edward words with a man who specializes in Flame Alchemy, and even though Rin isn't a Tamer he does know people who are and has need the summoning circles they use to call upon their demon familiars.
In Rin and Edwards' anime films, the two meet a person that they end up protecting from the people their supposed to serve, since they had been hurt enough by their sides and in the end their new friends help Ed and Rin save the day. As Julia Crichton used the Star of Milos to help Ed and Al save the valley, while the friendship Usamaro formed with Rin had gotten him to sacrifice his freedom for the people of Assiah. The 2003 version of the FMA anime also has Ed meeting and helping child who others view as the enemy because of what he is. A bit like how Usamaro used his powers to save Rin, (2003) Wrath used his own blood to help Al find Ed. Rin and Ed even though know scary strong women who have served as their teachers and allies in battle. For a long time they had surrounded themselves with people who are much older than them before they meet and befriend people who are close to their ages, who have fought along side them in battles with the Homunculus and demons. Just as there are people who hate Rin for being the son of Satan, Edward as met a few people who hate him for being a State Alchemist, because of the many bad things that Satan and the Amestris Military have done. Ed and Rin even have strong feelings for certain blond girls who have treated their wounds, are close with their younger brothers and worry about Rin and Ed when Shimei and Winry learn of the dangers they have faced.
Edrin is a rare anime/manga crossover ship, as most fans have Ed and Rin fight each other in amine crossover showdowns; before a small group of fans decided to ship them as lovers or friends. In a few fanfics that have the two meeting each other on friendly terms, it commonly has one of them being transported to the other's home dimension, in which has Ed and Rin working together, with the help of their younger brothers and friends, to help either of them get back home. All while helping the other to fight the battles and villains of the dimension they were brought to, on the way.
Even though the anime/manga crossover ship has a very small fanbase, it still has its supporters who believe that a few of their similar traits can have the two get along with each other.
- Their first anime series don't follow much of their manga's original storyline, the same with their anime animated films, until their second anime series came out. Fullmeatal Alchemist: Brotherhood, with only two episodes being of the anime version alone, and Kyoto Saga, which is mainly focused on the manga's Kyoto Impure King arc as well as serving as the anime's second season.
- Both of their manga/anime worlds require a type of circle to do alchemy or summon demon, as long as the sketch circle and the designs in the center of them aren't damaged in anyway.