No; they're not the same.  Allspice are unripe, dried berries from the tropical plant u003cemu003epimenta dioicau003c/emu003e (Jamaican pepper), while mixed spice is a blend of several different spices, including allspice.  u003ca href=u0022https://www.cookingfantastic.co.uk/ingredients/31-allspice-v-mixed-spiceu0022u003eThis chartu003c/au003e contrasts the differences.