What means character here:
A lovely old house, full of character.
The house full of what exactly?

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It's not like any other house. It is full of little things that make it different and interesting.

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This is an example of "personification". Human qualities are attributed to a house so, like a human, the house can be full of personality (meaning bursting with personality).

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It means full of distinctive qualities. Something unique or that stands out.