My 16mth old thinks she's 4... HELP

[Kitty74]

My very lovely and usually content 16 mth old dd has a 4 year old sister whom she worships. Now that she is walking and talking a little she thinks that she can do /have everything that dd1 does/has. When she doesn't get it we are starting to get tantrums and dd1 gets annoyed (rightly so) that her game has been interupted.
I really do not want to go down the buying 2 of everything route.
Has anyone else had this problem?
How did you deal with it?

DD2 get s presents for bday etc that DD1 already has.
For hr second bday last month, DD2 got a dressing table like DD1's and a doll like hers.
For Xmas I am seriously thinking about asking for identical presents.