Halliday & Matthiessen (2014: 289):
If all ‘possessive’ and ‘circumstantial’ clauses had the intensive verb be as the Process, we could perhaps interpret them as subtypes of ‘intensive’ clauses – subtypes where possessor and possessed are related to one another and where circumstances are related to one another. Here the piano is Emily’s would be like our earlier example her name is Alice, where naming is construed as an aspect of one of the participants (cf. also the owner of the Piano is Emily). Under this interpretation a clause such as Emily has a piano would be the odd one out because here the sense of possession is construed in the process in the first instance (the verb have), not in (one of) the participants.
However, Emily has a piano is not the odd one out; it exemplifies a regular option throughout for all ‘possessive’ and ‘circumstantial’ clauses. This is the option of construing possession or circumstantiation as process. Thus alongside the piano is Emily’s, we have the piano is owned by Emily; and alongside Emily is like her mother, we have Emily resembles her mother. Here own means ‘be + possession’ and resemble means ‘be + like’.