I'm not a food scientist but I have many years of experience in both commercial and non-commercial food handling and preparation and, as far as I know, I haven't made anyone sick from eating my food.
Here's my take on the matter. Any time you're working with raw food, particularly meats, poultry, eggs or seafood a little red warning light should come on in your head telling you to use extreme caution in food handling. In this instance, you're dealing with two different high-risk raw foods: eggs and fish. Additionally, while you're working with them, they are unrefrigerated which creates an additional risk factor.
The moment you dip the first piece of fish into the egg wash, you have introduced the risk of cross-contamination. Any contamination (bacteria or parasite) will continue to live and multiply when you re-refrigerate the egg wash, although growth is slowed at temteratures below 40F. When you use the egg wash the next day you have a potential cocktail of contaminated product. It doesn't matter that you are using it with more fish from the same package. You've already introduced the possible cross-contamination into the wash the first time you used it and have provided conditions to multiply the risk. We aren't talking about spoilage, so odor isn't any guide at all to safety.
When in doubt, throw it out! In this case, it should be discarded (or fed to the dog) immediately after use.
Egg wash used and reused for brushing on baked goods is a different, less-risky matter. However, I still wouldn't store it for re-use.
My guideline for raw eggs is that once the shell has been cracked, use it immediately and throw out any unused portion. Similarly, If I find an egg with a cracked shell when I open the carton, I throw it out.
Maybe I'm overly cautious, but you should be too. My position is supported by best practices and guidelines in both commercial and non-commercial food handling procedures.
"Iím going to break one of the rules of the trade here. Iím going to tell you some of the secrets of improvisation. Just remember ó itís always a good idea to follow the directions exactly the first time you try a recipe. But from then on, youíre on your own." - James Beard