Does "whatever" have similar meaning with "any"?

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Whatever is similar in some sentences, but is sometimes rude.

“What type of ice cream do you want?”
“Whatever type”
This is a bit rude

“What type of ice cream do you want?”
“Any type”
This is more polite

“You’re allowed to paint the house whatever colour you like” is the same as “you’re allowed to paint the house any colour you like”

Finally, whatever just does not work sometimes.
“Do any of you like dogs?” Is good

“Do whatever of you like dogs?” Is wrong

I hope this helps :)