What the two graphs look like
Yoast SEO prints an @graph whose Organization node uses home_url() + '#organization' — typically https://example.com/#organization — plus related fragments such as #website, #logo, and #webpage. The script tag is usually classed yoast-schema-graph.
Rank Math prints its own graph. Newer builds identify nodes with #schema/ patterns (for example /#schema/Organization) and a rank-math-schema script class. Even when the visible name matches, the @id is a different entity as far as a consumer is concerned.
Collision, not a ranking penalty
Two Organization nodes on one homepage is a collision: the consumer has to pick a story. That is not usually a Google ranking penalty. It can still weaken rich-result eligibility. We cite the job, not traffic stats. If both plugins are active with schema output on, a homepage often carries two Organization nodes, two WebSite nodes, and mismatched logos if each plugin was given a different image.
Pick one emitter per type
Disable Organization and WebSite in the plugin you are not using as the source of truth. A theme can add a third copy: JSON-LD without those plugin classes, or microdata that GraphClash will not see (we only read application/ld+json). WooCommerce can add Product (and sometimes extra Organization or WebSite) on shop URLs. GraphClash detects this on a public URL. It does not log into WordPress or change settings.