Partner is trying to Pin an action in sales order. For example, when they pin -Orders and all the actions below, after they close the page and they come back to it "Orders" is not pinned but "Process" is. They want when they pin order and close the page, to stay the same after they open it again.

This is by design. If the partner hovers over the "Pin" icon the caption shows "Pin action bar". This feature pins the Action Bar in place. It does not pin the chosen Action Menu, i.e. Order in this scenario.
So when he closes and re-opens the Sales Order he can see the Action Bar is indeed pinned in place. However, the default Action is highlighted, i.e. Process.

They are pinning the action bar to the page, they are not pinning the specific action bar to the page. The Process Action bar is the default and you don't change that by pinning an Action bar to the page. Pinning just means the whatever Action bar is active will always be open on the page. The Active Action bar is not saved, it is selected.
They want the ability to ping a specific action bar to the page.