[sirinsidiator]

ZO_Tooltips_ShowTextTooltip also just calls AddLine:

function ZO_Tooltips_ShowTextTooltip(control, side, ...)
    if side == nil then
        InitializeTooltip(InformationTooltip)
        ZO_Tooltips_SetupDynamicTooltipAnchors(InformationTooltip, control)
    else
        InitializeTooltip(InformationTooltip, control, SIDE_TO_TOOLTIP_SIDE[side], OFFSETS_X[side], OFFSETS_Y[side])
    end
 
    for i = 1, select("#", ...) do
        local line = select(i, ...)
        InformationTooltip:AddLine(line, "", ZO_TOOLTIP_DEFAULT_COLOR:UnpackRGB())
    end
end

The last parameter of AddLine is called setToFullSize:

AddLine(string text, string font, number r, number g, number b, AnchorPosition lineAnchor, ModifyTextType modifyTextType, TextAlignment textAlignment, bool setToFullSize)

My guess is, that you need to add an element that is wider than your text in order for text alignment to work when you have setToFullSize set to false. Try adding a line of whitespaces.
You could also try to set a width constraint and setToFullSize, but you should only do that if you have created the tooltip control yourself (never on the built in InformationTooltip and others).