Ash and Lillie are best friends and classmates at the Pokemon School. Although Lillie sees Ash as a friend, she has shown to have some sort of romantic interest in Ash.
Ash (along with Kiawe, Sophocles, Mallow, and Lana) has always been supportive for Lillie when she had her fear of touching Pokemon and wanted to conquer it, and had made great progress. When Lillie temporarily went back to square one and couldn't touch any Pokemon (including Snowy) after remembering Type: Null in a memory from the past of the incident that caused Lillie to develop her fear of touching Pokemon, Ash and the others felt bad for Lillie and encouraged Lillie to never give up and keep trying. At the end of the episode, Lillie fully conquered her fear and can touch any Pokemon, and Ash and the others were so proud of her. And Lillie (along with the others) has always been supportive for Ash during his battles, Trials, Grand Trials, and his new Pokemon team members.
Ash and Lillie (along with their classmates) are a part of Professor Kukui's class at the Pokemon School, and are also members of the Ultra Guardians: Ash is the blue Ultra Guardian and Lillie is the pink Ultra Guardian.
They also have always been there for each other when they were sad and down on their luck, such as in "SM98" after Ash and Rotom got into a fight, Lillie talked with Ash and told him it was going to be ok.
Before leaving Alola in "SM146", the series finale, Lillie had a flashback to some of the moments she shared with Ash, and told him "thank you", and that's she's glad she met him. Ash returns this sentiment and waves goodbye to her from the dock.
This ship is one of the most popular shippings between Ash and a female companion. AureliaShipping has received a large amount of supporters (along with AlohaShipping and BlueAngelShipping) since the Sun and Moon Anime debuted. Ash and Lillie are very close friends, as said previously Lillie may have a sort of romantic interest on Ash, but we can't guarantee that. Despite this, the ship still continues to thrive over the course of the anime.