Asset share is basically "how much money would be in this product's bank account after receiving all the premiums, paying out all the expenses and claims, and earning investment returns". (Cost of cover is basically the cost of claims paid).
So the asset share are the assets of the policy - what is available to pay out surrender values. Setting surrender values needs to consider that - it is bad practice (but sometimes unavoidable) to pay out more to any policyholder than their portion of the asset share. Think of a situation where you have to discontinue all policies in a book - you would have to pay out more money than you have, and we try to avoid that.