As has been said, it seems to stem from nothing of the kind.
This is normal. It's not a question of tradition so much as of practicality: if you've got to keep two complete sets of silverware, dishes, and so on absolutely and rigidly separate, you label them very carefully indeed. If you ever put meat on a dairy plate, you (a) destroy the meat, and (b) destroy the plate. If you're deeply, deeply wedded to the plate, there are ways to purify it, but they're difficult and often expensive. Example: if you convert your kitchen to kosher in the Orthodox Ashkenazi tradition, the nice men will come to your house and run a blowtorch over the entire interior surface of your oven, just for starters. I believe it's usual to throw away all your dishes, silverware, pots, and all that when converting the kitchen, but I'm not really sure.
A good general prescription, but it doesn't make much difference here. If this kitchen is pareve and they're serious about it, it means nothing, but nothing, ever enters that kitchen that isn't pareve. I wouldn't be surprised if they preferred that you not bring your own knives, and since you'd have to get a new set anyway, you're probably best off using theirs. If nothing enters the kitchen non-pareve, and you obey whatever initial hygienic practices they deem appropriate, then nothing can depart the kitchen non-pareve. So this is easy.
Actually, any professional kosher kitchen is easy. If the kitchen does both meat and dairy, it does them on different days, or else is divided cleanly and clearly in half (or there are two kitchens). You just don't cross the line with any utensil of any sort, and everything's clean. Some pro kosher kitchens do have a joint space where pareve items are kept, but it's more usual to have two: it's just easier not to worry whether a cook is going to grab for sugar when his hands have a bit of butter on them. So long as you obey the strict division, so that today you're on meat and never never never dairy, there's nothing to worry about.
Running a pro kosher kitchen is quite another matter.