[Dashboard] embeddable filters are not displayed

[mbondyra]

Kibana version:
main

While working on #186642 we noticed that the filter icon doesn't display correctly for Lens embeddable. After digging more, it looks like the problem appears for all the embeddables.

expected display

actual display (when loaded for the first time)

(For Lens, it works on main because it hasn't been rewritten yet, but looks like the root cause might be the same as for visualize.

As for Maps, it works correctly, but once you click on the options for "maps by reference + filters" map vis, and Edit map, the filter is lost (video documenting it below)

Sep-20-2024.09-54-51.mov

Also, side question - did we remove the 'linked to library' icon (see the expected display) or is this a bug too that it doesn't display? 🤔

To reproduce you can use the SOs from here. Change the extension to .ndjson: filtersbug.json

[elasticmachine]

Pinging @elastic/kibana-presentation (Team:Presentation)

[teresaalvarezsoler]

Afaik, the Saved to library icon should appear too. I think we will remove it as part of this other PR but it hasn't been merged yet #182535. Can you please confirm @andreadelrio?

[nreese]

Afaik, the Saved to library icon should appear too. I think we will remove it as part of this other PR but it hasn't been merged yet #182535.

Library notification was removed outside of hover actions in #190797.

[nreese]

This issue covers a few items. I am going to close and break items out into separate issues.

Created #193598 for visualize embeddable not displaying filters notification.

Created #193601 for the maps filter badge disappearing.