Is the amount of an item you give affects the rate of friendship with the villagers?

it does. the more rare/expensive the thing you give them is, the more it’ll boost your friendship with them
 
I doubt it. Each villager has their favourite colours/styles. The royal crown will only match the style/colour of certain villagers. The rest will probably say 'this is not what I usually go for - but thanks!' (my source is the official companion guide)

It does increase villager friendship however if you wrap the gifts in gift wrap.
 
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