Can I add a share button for a site that isn't in the presets?

Yes. In Marketing → Social Sharing, click + Add Share Button and fill the fields in yourself instead of using Quick add:

• Name: what the button is called
• Icon Class: the icon to show, in the same style as the presets (for example fa fa-facebook)
• Share URL: that site's share address, with %%URL%% where the product link should go

Click Add Button. Open a product on your store afterwards and test the button.