I'm not really sure if these words carry the deep meaning you think they do.
相性 means "compatibility"
家庭 means "household"
You also shouldn't put them together like 相性家庭 because it's kind of nonsense. Would be like writing "compatibilityhousehold" on your arm.
The way they've been drawn in that picture seems a bit strange too. Especially 相 and 庭. Maybe that's some special caligraphic way though?