Ryuu no Haisha

Both of these anime: - Have plots that revolve around the human race's symbiotic relationship with giant legendary creatures who are used as ultra powerful weapons. - Have grandiose endings set on a huge scale with lots of death - Are thematically similar e.g. fear of death, betrayal, war etc. - Have main characters much younger than their comrades who are yet to truly mature - Have the celebrated Hideaki Anno deeply involved in both projects 

Both The Dragon Dentist and Suisei no Gargantia have a male and female protagonist and the plot is similar in the beginning of a soldier arriving on a ship that slowly becomes one with the community. 

Hisone to Maso-tan and Ryuu no Haisha, for starters, both focus on humans and dragons coexisting with each other and what that dynamic entails in their respective stories. While both shows have a healthy dose of action elements, if you like anime with a chill atmosphere, they both have it in ample sums as well. Moreover, if you enjoyed the candid, "daily routine" elements either one established, the other could be up your alley. 

A fantasy world in which people have special powers with assigned tasks. 

A fantasy world colliding with the real world. A war and a creature with magic power takes part. 

The style of Ryuu no Haisha certainly reminds one the old Ghibli movies. It's based on some old Asian legends and myths but somehow modified to the nowadays or recent world and society. While in Laputa we have the flying castle, in Ryuu no Haisha we have the dragon large as a castle. There are people involved seeking it's power for their own purposes and there are also people trying to protect it. 

Similar concepts of verticality and prejudice. Similar male/female protagonist duo. (coming from opposing groups, one saves the other and they become close) 

Both shows follow characters who are suddenly thrown in a fantastic world unknown to them (and the viewer), where they have to work and figure out the rules as they grow.