Given that marriage was (and in many cases is) a religious institution, the "wife" is viewed as the property of the man. This is why the more traditional phrasing is "man and wife," rather than "husband and wife." Given that the wife, and the subsequent children, were all viewed as property of the husband, the substitution of the woman's last name for that of her husband can be viewed as a simple transferal of property from the father to the new husband.
TC makes many excellent points above, most notably the fact that the woman is denied a coherent recognizable individual identity. This is naturally due to the fact that the woman was viewed as property and hence devoid of individual identity apart from that of her owner, her husband.
As for the OP's question on marriage, I am a male and hence the question as stated doesn't apply to me. I was merely hoping to contribute to the discussion on names and patriarchy.
- August
If we have no business with the construction of the future or with organizing it for all time, there can still be no doubt about the task confronting us at present: the ruthless criticism of the existing order, ruthless in that it will shrink neither from its own discoveries, nor from conflict with the powers that be.
- Karl Marx