Short answer
There is no universal egg substitute. Choose according to what the egg is doing in that particular recipe, and use a tested recipe when eggs supply several functions at once.
For binding or moisture
Hydrated ground flax, chia, starch gels, yoghurt and fruit purées can work in some cookies, muffins and quick breads. Each changes flavour, colour, water content or density. A replacement that keeps a patty together may not lift a cake.
For foaming or emulsifying
Aquafaba can form a foam for some meringues, while mustard, lecithin and formulated products can help emulsify. These substitutes behave differently under heat and may not provide the same browning or set structure.
When substitution becomes difficult
A recipe containing several eggs is often structurally dependent on them. Cakes, soufflés, custards and enriched doughs may use eggs for lift, setting, colour, richness and water at once. A recipe developed to be egg-free is more reliable than multiple improvised swaps.