This is silly. You guys are arguing extremes when the truth is likely in between. Know Face Man is at least partially right, as a huge % of people in the Free Cities (and a decent % beyond) will have heard of Aegon the Conqueror and his dynasty and know that they had the last dragons. Almost literally everyone in Essos has heard of Valyria so yes, they will associate dragons with Valryia. Many people will have heard of both.
Ultimately the point that an egg might give young Aegon away is a good one, as it's *possible* and not a particularly unlikely, either. So it's *possible* Illyrio considered this and didn't give him the egg. Also, the egg is massively valuable, so you don't want Aegon being a target for someone who wants to kill him and his protectors in order to take it. If you're going to give him an egg, there's no rush. Obviously Illyrio may not see it that way, but it's not a stretch at all.