We have three large black basins outside of our basement in which one appears to have a sump pump. The other two just have standing water. When my daughter turns on her shower there is a hole in the side of the PVC pipe that starts shooting out water into this sump pump basin. I’m not plumber but this seems like a HUGE issue if the house feeds into this. Am I correct? Would a plumber be the person to call for this or someone else.

if your daughters shower and ONLY the shower is piped to sump pump type basin, i wouldn't worry about it. it is only grey water and will do no harm.
besides, if you wanted to re-direct the shower drain into your main drain, it probably would be a big job???

If you're in a rural area with a well and septic, this setup keeps the shower water out of the septic system. If the barrel it drains to has a sump pump in the bottom, the previous owner was probably recycling the gray water to irrigate a garden or similar. Gray water can stink worse than black (toilet) water, so I wouldn't want this setup if I was in a subdivision with close neighbors, but out in the country this could actually be a good thing.

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Not a plumber by trade but a fierce DIYer

I just sold a house that had a gray water system for the showers and sinks. It is surprising that nobody else asked: Where is the OUTPUT of the sump pump? It is quite common and safe to pump gray water into a leach field or leach pit. If it pumps into a pond, stream or sewer it is likely violating various codes.