Engineering manager is someone who has a head count and is responsible for managing it (hiring, firing, giving feedback, promoting, assigning to teams, resolvin

Engineering manager is someone who has a head count and is responsible for managing it (hiring, firing, giving feedback, promoting, assigning to teams, resolving dramas etc.), while Product Manager is often a glorified analyst (depends on the relationship with the business). Given that, for managers, headcount is a way of assessing one's respectability, I'd say that EM has a higher change of going further up the ladder.