Story tabs shown in all blocks of the same environment

[RichDom2185]

• Create two blocks of code sharing the same environment
• Trigger a tab in one of them (e.g. showing a rune/some other module)
• Expected behavior: the tab only shows on the block you ran the code from
• Actual behavior: the tab shows up in both blocks

Likely requires not-so-trivial refactoring of StoriesSideContent.