Is there a word like "inherit" for when a widow(er) takes full ownership of the shared estate?

Inherit/inheritance seems to apply to heirs, successors, and new owners. Maybe new owner could apply, but it is strange to think of it that way, as it might suggest that the widow(er) didn't have a share in it until the spouse died.

Is there a better word than inheritance to describe the estate with respect to the surviving spouse? A better word than inherit to describe the widow(er) taking full ownership?

Yes. Such a widow who inherits her husband's estate is a "dowager." The inheritance is a "dower," and the verb is "to dower" (e.g., Dowager Jones' late husband dowered her the estate.)

vest transitive verb 1a :  to place or give into the possession or discretion of some person or authority;especially :  to give to a person a legally fixed immediate right of present or future enjoyment of (as an estate) —M-W

In a joint tenancy arrangement, unlike a tenants in common arrangement, ownership would transfer as described in question if one owner outlives the other.

Where the property is held by co-owners as joint tenants and one co-owner dies, the remaining share would vest in the survivor or survivors by the right of survivorship and would not devolve under the will or intestacy - until of course the death of the last survivor, who can leave the whole of the property to who he pleases as sole owner. - http://www.lawteacher.net/resources/land-law/land-law-coowner.php