Is there a way to control the URL rendering on Messages App on iOS 10 with special meta tags, such as Open Graph tags or Twitter cards tags?
It seems like it is not based on apps installed, so I'd assume it is a white list made by Apple if we cannot control it.
Example:
Apple calls the metadata used for rendering web content throughout iOS "Web Markup" — this is largely based on schema.org and Open Graph. For details see the Mark Up Web Content section in Apple's App Search Programming Guide.
You can validate your metadata using the API Validation Tool or using iMessage in iOS 10.