Definition of Done, one for organisation and one for product?

The Definition of Done is specific to a product. It helps all stakeholders, on and off the Scrum Team, understand what it means when a Product Backlog Item is done and an Increment is made available. In scaled Scrum situations, the expectation is that all Scrum Teams working on a product "comply with the same Definition of Done".

I think it's important to define what it means to "comply with the same Definition of Done". In a scaled Scrum environment with multiple teams working on the same product, this doesn't mean teams can't have their own Definition of Done. It does mean that no team can have a less strict Definition of Done than another team working on the same product. In fact, experimenting with the Definition of Done is best done at the team level, and effective practices can be shared with other teams.

The same concept can be applied at other levels. If you have a closely related suite or portfolio of products, some elements of the Definition of Done may be shared across all of those products. An organization may have key elements of a Definition of Done that apply to all product portfolios or products it creates and maintains.

I would offer some caution, though, and recommend considering the absolute minimum required content for higher levels of the Definition of Done. You don't want to be in a position where a team wants to remove something because it's not necessary or would create a burden for their particular case. A Definition of Done that flows down to a team should be as minimal as possible to ensure the desired level of quality.

The Definition of Done applies to the Product. It is a commitment the Developers make to assure the quality of each Product Increment.

The parent organization should nevertheless be the first port of call for a Definition of Done, because corporate standards ought to be respected as a minimum. Company brand and reputation should not be put at risk.