[SDPhantom]

What you have in the image isn't the achievement frame, it's a tooltip. Specifically ItemRefTooltip.

function ShowCustomTooltip(title,character,date,description)
--  Mimics SetItemRef() starting at ItemRef.lua:389
    ShowUIPanel(ItemRefTooltip);
    if not ItemRefTooltip:IsShown() then ItemRefTooltip:SetOwner(UIParent, "ANCHOR_PRESERVE"); end
 
--  Mimics :ItemRefSetHyperlink() at ItemRef.lua:791
    ItemRefTooltip:SetPadding(0,0);
 
--  Custom Text (Replaces :SetHyperlink() at ItemRef.lua:793)
    ItemRefTooltip:ClearLines();
    ItemRefTooltip:AddLine(title,1,1,1);
    ItemRefTooltip:AddLine(" ");
    ItemRefTooltip:AddLine(("Achievement earned by %s on %s"):format(character,date),0,1,0);
    if description then
        ItemRefTooltip:AddLine(" ");
        ItemRefTooltip:AddLine(description,1,1,1);
    end
 
    if ItemRefTooltipTextLeft1 and ItemRefTooltipTextLeft1:GetRight()-ItemRefTooltip.CloseButton:GetLeft()>0 then
        ItemRefTooltip:SetPadding(16,0);
    end
end

As for sending this through chat to another player, links are encoded with an achievement ID and the GUID of the player. Links of any type do not support custom data. You'll have to find your own way to encode and transmit this info and it'll require your addon on both ends.