[merlight]

Found it, the bar is re-anchored every time the skills window is shown/hidden:

-- ingame/scenes/ingamefragments.lua
 
function ZO_SkillsActionBarFragment:Show()
    ZO_ActionBar1:ClearAnchors()
    ZO_ActionBar1:SetAnchor(BOTTOM, ZO_Skills, BOTTOM, -40, 40)
 
    ActionButton9:SetHidden(true)
    ZO_ActionBar1KeybindBG:SetHidden(true)
 
    ZO_FadeSceneFragment.Show(self)
end
 
function ZO_SkillsActionBarFragment:OnStateChange(oldState, newState)
    if newState == SCENE_FRAGMENT_HIDDEN then
        ZO_ActionBar1:ClearAnchors()
        ZO_ActionBar1:SetAnchor(BOTTOM, GuiRoot, BOTTOM, 0, 0)
        ActionButton9:SetHidden(false)
        ZO_ActionBar1KeybindBG:SetHidden(false)
    end
end

You could either post-hook these two functions on SKILLS_ACTION_BAR_FRAGMENT, or replace that fragment with your own modified version. Haven't tested it, but looks a bit cleaner than hooking functions, so I hope it will work:

local customSkillsActionBarFragment = ZO_SkillsActionBarFragment:New()
 
function customSkillsActionBarFragment:OnShown()
    -- call original (class) method
    ZO_SkillsActionBarFragment.OnShown(self)
    -- now do your stuff
    ...
end
 
function customSkillsActionBarFragment:OnStateChange(oldState, newState)
    -- call original (class) method
    ZO_SkillsActionBarFragment.OnStateChange(self, oldState, newState)
    -- now do your stuff
    ...
end
 
local skillsScene = SCENE_MANAGER:GetScene("skills")
skillsScene:RemoveFragment(SKILLS_ACTION_BAR_FRAGMENT)
skillsScene:AddFragment(customSkillsActionBarFragment)