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It seems that the order number of a MIP variable can be obtained from its string representation as follows:

def get_order_number(mip_var):
    return int(str(mip_var)[2:])

It seems that the order number (backend index) of a MIP variable can be obtained from its string representation as follows:

def get_order_number(mip_var):
get_backend_index(mip_var):
    return int(str(mip_var)[2:])

I've added a request for an "official" way to get this index at https://github.com/sagemath/sage/issues/38799