The reality is you will never use a common secret storage anyway as usually and even when using multi-cloud tools like terraform you write a lot of cloud vendor specific stuff because they do not share the same provider. So using different secret storage API is not a huge deal and pretty much a moot point.

Also my vision of multi-cloud in large org hasn't been that a particular product/app or team was ever using multiple clouds. My experience is that large orgs like to have multi-cloud support because they grow by acquiring other companies regardless of which cloud vendor they are using so you just want to provide standards and templates for everyone. Obviously said templates will be usually cloud vendor specifics.