What does a product manager do at a gaming company?
Whatever the company wants, could be schedule monkey, could be community outreach, could be designer. Only way to be sure is to talk to hiring manager.
Is the experience as a gaming product guy taken seriously by traditional product companies? I worry if this will be considered frivolous stuff
You will have to try harder yes unless you work on a large project with demonstrable skills and results.
Dogfood; talk to gamer community without calling them incels etc.