It's true that BNF grammars can define non-regular languages, and so can't be parsed with an FSM. I wouldn't expect Leslie Lamport to have overlooked that fact, though... Skimming, I think what's being described are infinite state machines. FTA:
"For example, a BNF grammar can be described by a state machine whose states are sequences of terminals and/or non-terminals. The set of initial states contains only the sequence consisting of the single starting non-terminal. The next-state relation is defined to contain <s, t> iff s can be transformed to t by applying a production rule to expand a single non-terminal."
"For example, a BNF grammar can be described by a state machine whose states are sequences of terminals and/or non-terminals. The set of initial states contains only the sequence consisting of the single starting non-terminal. The next-state relation is defined to contain <s, t> iff s can be transformed to t by applying a production rule to expand a single non-terminal."