When I have crocheted projects in the past, I have used all of the same lot number for the yarn even though all looked the same to me. When is using the same lot number really important? Is it only with natural fibers or natural fiber blends?
Yarns with the same dye lot number were all dyed at the same time. Because dying is a chemical reaction, many different factors can make items dyed at different times turn out differently, even if the dye "recipe" was identical from one dyeing session to the next. The temperature could be different, there could be slightly more or less pigment in a teaspoon of the dye powder, the water used for the dyes could have more or less chlorine from the water treatment plant, the fiber being dyed could have been scoured with a different type of soap, and any residual elements could interact with the dye. There is tons of information about this on the interwebs-here is one article from Wikipedia.
All that is to say that it is almost impossible to get an exact match between yarns dyed in different dyeing sessions. It doesn't really matter what the base fiber is, or whether the dyer is an independent dyer working in their own home or a major yarn manufacturer--there will almost always be a slight variation of color between yarns dyed in different dye lots.
So, if you are going to be making something where you have a lot of a single color in a block, you usually want to purchase yarn from the same dye lot, otherwise you will probably see a slight (and sometimes not so slight) color demarcation where you have changed yarn. In this picture, someone had to use yarn from a different dyelot to finish the sleeve:
BTW--if you ever find yourself in a situation where you do have yarns from different dye lots, the best way to minimize the issues with color mismatch are to use the yarns from both dyelots at the same time--knit/crochet two rows with one yarn, and then switch to the other yarn. Carry the unused yarn loosely up the side of the piece until you are ready to use it again. By limiting the area of one color to a small section, your eyes will make the colors blend together and you are much less likely to notice a distinct color change.
This is a very good question!
Commercial-sized lot dyeing and computerized color matching have lessened the chance of noticeable lot variation visible to the human eye, which doesn't mean there isn't some color variation. All manufacturing processes have "acceptable tolerances" which means they don't necessarily hit the nail on the head each time, but that the variation is deemed "acceptable."
In most cases, there's less concern about lot-to-lot variation with the big commercial yarn manufacturers such as Bernat or Lion, but with some craft store brands or "off brands" I'd be careful. Each manufacturer or brand sets their own acceptable color variation tolerances.
Another watch-out for off-brand yarn is that small resellers can overdye a color that didn't sell, or wasn't up to specifications, or for numerous other reasons. Over-dyed yarn is something you want to avoid since it has a different color "heritage."
It's still a good idea to check lot numbers for things that matter - I wouldn't be so worried about lot numbers if I were making children's toys, but would be more careful when making a sweater or blanket - even slight "technical" shade differences may be evident to some people under different light sources.
You are correct that natural fibers have inherent differences because of animal-to-animal or plant-to-plant variation due to genetics or weather or dozens of other variables. For commercial natural yarns, the above comments about acceptable tolerances in color matching still hold. But for small-lot "artisanal" dyeing, lot number matching is essential.
It's worth noting that some synthetic yarns and fibers, e.g., acrylics, can be dyed at the slurry stage, in other words, before the fiber or yarn is even formed. This means the color is inherent to the structure, not something added. These solution-dyed yarns are often used for commercial purposes such as indoor-outdoor fabrics that are expected to withstand excessive wear and exposure to the elements. The color matching information above still holds, it's just an interesting factoid.