I would use schema.org Organization markup on both of the domains, but it would be good to connect them together by 'parentOrganization' and 'subOrganization' properties.
Here is how:
If you have a blog on subdomain, it is good to add a link to your "NewsMediaOrganization" on your main website:
...
"@type": "Organization",
...
"subOrganization": [
{
"@type": "NewsMediaOrganization",
"@id": "...unique ID... - usually homepage URL with #someslug",
"name": "your business name"
},
...
Then on the blog subdomain, use:
...
"@type": "NewsMediaOrganization",
....
"parentOrganization": {
"@id": "...unique ID - usually homepage URL with #someslug",
"name": "Yourbusinessname Blog"
},
....
If you register your blog on Google News Publisher, and write some good content, this will increase your chances in getting some love from Google on Google Discover.