What Morimoto Shigeki (game designer and software developer for GAMEFREAK) said when asked why Lugia is not Water-type, quoted from Nintendo DREAM November 2009 issue.
To start, the first type was decided to be Flying-type since Lugia have wings. At the time Psychic-type was the symbol of powerful Pokemon. As we wanted Lugia to give the impression of being a powerful Pokemon, we settled on Psychic-type for the second type, instead of Water-type.
In short, the reason why Lugia is not a Water-type despite being underwater, is due to its Flying-type-like appearance, and Psychic-type being the symbol of a powerful Pokemon.
Legendary duos represent opposite concepts, for example Groudon and Kyogre represent land and water, while Dialga and Palkia represent time and space. In the case of Ho-oh and Lugia, although they do not have a direct connection in the story, we imagine that they represent the sun and the moon, day and night, yin and yang (positive and negative), bottom of the sea and the sky, above and below.
Ho-oh and Lugia were born in the world of Pokemon Gold and Silver, for the new feature, time system, gave us day and night cycle. Thus, they represent the sun and the moon, day and night.
Basically, the reason why Lugia isn't Water/Flying type is only because of circumstance. At the time, the Psychic-type was symbolism for a powerful Pokémon, (if you didn't know, Lugia was very powerful at the time it was released) and the reason why it's also Flying type is well, because it has wings and quoting Morimoto, "has a Flying-type like appearance". Ho-oh also represents the Sun, and therefore Lugia is the Moon. (now I wonder why they didn't make Lugia a Dark type, but no more questions, and anyways, Psychic has at least some representation as the moon)
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