A lot of people like to tell horror stories of epoxy grout and over the years its uses has become a bit cloudy. However, epoxide resin based grouts are specifically formulated to resist chemical attack and to provide a durable, abrasion resistant waterproof finish that is hygienic and impervious.
The cured grout does not transfer taints to foodstuffs, permit the entry of bacteria, or dirt and is easily maintained in a sterile condition. With this in mind it makes a perfect food counter grout.
As for use in a shower base it is perfectly acceptable to use and in some cases preferred over cement based grouts as it does not allow water to penetrate.
As for staining and discoloring these will happen in a lot of situations it really depends on the quality of your water and what minerals it has in it. Over time they will tend to leech on any surface that has water coming into contact with it. Even plastic nozzles will do it. However most surface cleaners will remove such stains very easily as long as they are not left for extended periods to build up. I should also point out these stains wont appear over night. They can take months and or years so its not really something concern yourself with hugely.
I'm very sorry to say this but you're going to have to remove all the tiles and start over. You will never been able to prevent this from leaking fully.
It may also mean removing and replacing the backing as well.
The critical step you missed was the waterproofing membrane that goes under the tiles and in and over the drain pipe. The tiles don't stop the water at all but are for aesthetics only (and to protect the water proofing). What makes the shower water tight is a sturdy backing covered with a water proofing membrane like liquid flash (one I used) when I did my tile shower. http://www.constructionchemicals.com.au/products/waterproofing-membranes/
But there are many different brands. They are a paint on product and very easy to install.
Don't expect your tiles to be waterproof. At best they are only splash resistant. Water goes right through the grout and can also enter around the side of the tile where the grout is. Porcelain tiles aren't quite as porous as ceramic, but most wall tiles are ceramic because they are a lot lighter. These tiles act like a wick and just draw the water through to the back of them. If you don't fix this you could be in for some serious mould issues down the track.
Unfortunately there is no simple fix to this but to start over. I would not recommend just trying to seal it. It's like covering up rotten wood with paint, it won't last and won't fix the problem long term. Although you've got water resistant plasterboard. It won't last long term. These plasterboards are just designed to repel water to keep it out of the gypsum they are made of, not designed to be constantly wet.
Interesting construction that your builder used water treated plasterboard + chipboard for the construction. 6mm or 1/4 in thick concrete based tile & slate underlay board + waterproofing membrane is all you would have needed. I wouldn't recommend tile on plasterboard at all. Even with the layers that he has. It's not as strong.
Since I know how much work goes into these DIY tile showers I will say this. If you feel overwhelmed, call in an expert. They will have the job done in a week and you'll be able to move on with life. But if you're up for the challenge and want to learn to do it right this time, then dive in and go for it!
Best Answer
It is usually always better to use like materials when fixing something. If you use regular concrete based grout to patch a hole in epoxy grout, not only can you not be sure of a texture and color match, there is the risk that different expansion rates will simply cause the concrete to crack.
Not to mention, concrete grout is not stain proof, nor water proof. Epoxy grout is both. So eventually the difference will show...