So this leads me to the 3rd element of the triangular theory. The mechanics you chose must have some relations with the theme in order to generate the right feeling you want the players to experience by playing the game. So the third element in the formula is the feeling which could also be the "experience".
Of course, video games are better at generating feeling and experience since here it's a full audio video media. But still, there are ways to generate a certain amount of feeling in a game. For example: in a horror game, you might not be able to scare people, but you might have mechanics that can stimulate tension, risk, discovery of a puzzle, etc.
The 3 elements of board game design could be illustrated by the triangle in the picture.
The theme of the game requires a certain feeling. For exemple: In a horror game, you could want people to feel tension.
To generate that feeling, you need a mechanic. For exemple: If you want to generate tension you could implement a time limit in number of turns. Or even use sand timers.
Finally you need to explain this mechanic with a theme. For example: The time limit is explained by the day and night cycles like in fury of dracula.
So here you have a perfect cycle where you have a mechanic, where the theme is explain it. But the feeling generated by the mechanic match with the theme. So tha the theme does not seem pasted on the mechanic.
When designing a game following the theory above, you can now approach the design from 3 different ways. So you can now either approach the game either by theme, mechanic or feeling and then define the two other missing elements.
Of course this theory of design is pretty abstract. You never detail all your mechanics to know if they actually all fit. It is more a general guide line that you should follow and keep in mind when you designing a game.