Dr. Rin ni Kiitemite!

Although the plots of these two series are almost completely different, and Dr. Rin is darker and more complex, they have a similar "vibe," probably because they are both shoujo titles animated by the same studio and share some staff. 

both have secret hero identities, both have some sort of power to destroy evil spirits, both have romance, basically has the same plot, if you like one then you will definitely like the other 

-Same story about four Gods -Same love story about Priestess and the Warrior. Example: Tamahome and Miaka, Meirin and Yuuki Asuka -The main protagonist is a priestess. Same plot story, and romance.  

Both anime are heavily influenced with fortune telling and "fate." Both main characters can predict fortunes and the future. There's romance in both and a huge fight that involves fortune telling. Dr. Rin is more humor-based while Escaflowne is a bit more of action drama. While both anime have a clear story to tell, Escaflowne is more serious and complete. 

Both series are magical girl series with a unique theme (weddings and feng shui), with similar animation and with the love interest of the main girl being members of their school's soccer team.