When I was a kid, my family lived in Bavaria for a while (South Germany). My dad is an agronomist and is originally from South Italy (so he grew up with lemon trees), and when my parents tried growing a lemon plant, with all the amenities of a 20th century home (like constant heating, modern fertilizers, my father's advanced knowledge of fruit growing), they had mixed success. It was very time and care intensive , and the plants would all wither within the year (and way before the point at which they could be called "trees").

I find it hard to believe that, save some kind of magic, a lemon tree could grow in the Essosi equivalent of the Pianura Padana (where I live currently), to the point that it becomes a tree, in the open next to a door.

Either that plant was brought over already grown and withered very quickly, or it wasn't Braavos.