'Bird egg' would be a compound noun. With compound nouns are we have two uninflected nouns put together: xy. (Uninflected in that we have no genitive 's malarkey or suchlike).
'y' is what the thing is.
'x' tells us kind of 'y' we're talking about.
So 'duck egg' or 'ostrich egg' are more specific than simply 'egg', in that they specify what type of egg it is. Likewise 'birdbath' and 'bird table' specify that these are types of baths and tables used for birds. These slightly unusual objects need to have 'names', to distinguish them from more 'ordinary' eggs, baths and tables, so we invent compound nouns to name them.
Most of the time when we are talking about eggs in general, we expect them to come from birds, so we don't need to create a special compound noun 'bird egg'. It almost goes without saying - at least in a culinary context - that an egg will be of avian origin. In the rare cases where we have to distinguish between a 'regular' egg and a crocodile egg, for example, we have no handy compound noun at our disposal, so we resort to the slightly clumsier genitive form, 'bird's egg'.
That's my poultry ... I mean paltry ... two penn'orth, anyway.