I also homebrew, where using a "no-rinse" cleaner is a must - example. I haven't had any issues with using it for cleaning all the kombucha things, so I think its safe for the SCOBY and all that. I just clean everything with soap and water and then the last step is this no-rinse.
I also run the beer bottles through the dishwasher to clean them, no-rinse to sanitize
You don’t have to but I do. Sanitizing the containers minimizes the risk of germs or bad yeasts (which will change the flavor of the wine, usually for the worst). I use this sanitizer
LPT: Go on Amazon and order this because it actually works and isn't a remedy your grandma came up with: http://www.amazon.com/Strange-Brew-One-Step-oz/dp/B0064O7VYY/ref=sr_1_3?ie=UTF8&qid=1444977848&sr=8-3&keywords=one+step
Source: Brewing beer is messy and shit needs to be cleaned effectively.
I had a Stanley thermos that had the worst coating of coffee residue ever and 1 tsp and 15 minutes later, it was brand new inside.
I could wash the limes with the one-step stuff, right? https://www.amazon.com/dp/B0064O7VYY
I've used One Step for years. It's an oxygen-based no-rinse, but I rinse anyway.
TLDR: The "Full kit" looks like it has the basics. As others have said, you might want a food-grade plastic bucket for primary fermentation, and you'll need bottles/containers to store the end product in.
If it were a "complete" kit I'd probably put one together which included One step sanitizer to sanitize equipment,a plastic fermentation bucket, and an auto-syphon to make racking (i.e., transferring the liquid from container to container) easier.
If it's something you're interested in pursuing further, there's plenty more you could consider picking up. A bottle filler for the auto syphon, a filtration kit to help clarify wine/mead, fining products, you might want to look into picking up more things like yeast energizer and yeast nutrient (which it sounds like this kit comes with some) and sulfate/sorbate (to stabilize the mead before back-sweetening) etc.
There are lots of recipes and lots of help available, so read up and feel free to ask questions and have a lot of fun experimenting and trying new things :)