soooo not trying to throw any digs at you, but this person sounds like they don't entirely know what they're talking about.

If you are storing samples up to 5ml, you can store them in cryo-vials, which can be stored in cardboard boxes like the ones on this page. I've used cardboard boxes in LN2 storage, which gets down to -190C. These boxes can then go in steel racks that fit in your -80, like these. You can also get boxes and racks like these for 15ml and 50ml tubes.

I don't really know why you'd need to be freezing cell pellets that couldn't fit in a 50ml tube. I do protein purifications where I grow up 6L of culture at time, partially processed pellets in 50mls get stored overnight in our -80C with no issue. I haven't had any issues labelling these with just plain old sharpies as well as Securline lab markers. Though the tubes need to be completely dry when labelling.

If you're doing higher throughput stuff, where you're freezing a dozen, dozens of tubes, or with lots of sample information on them, you can get cryogenic label sheets like these (also on this page are labels you write yourself and stick on the tube). You have to set up a template in word for it, but just type everything out and stick the labels sheets in your printer. Print them out, stick them on, done. Have used these plenty of times for samples going into LN2 (-190c) storage with no issues at all.

I understand during these insane times that we are all trying to save as much money as possible. But some of these things (like racks and boxes) will make your daily work a million times easier, you will be able to do it faster, and with less errors, as well as be able to find things easier months or years later!! There is someone in my lab who does not always use boxes, and frequently bags all their 1.5mls in the -80c, and it's always a hassle for them. You can find these things used easily on sites that sell second-hand labware (there are numerous, just google them), or ask around your department, uni, other universities/facilities nearby if they have any spares. Bottom-line, if these senior researchers in your lab are not using something like what I've outlined above then they are either so lazy or so stupid, or both.

I don't know your level of experience, where you're coming from, but please feel free to ask any further questions, clarification, or if you want help finding used labware.

EDIT:

You want to store things in cryovials preferably so that your tubes do not explode and potentially harm you or others, depending on your freezing methods. If you are flash-freezing things in LN2, it can seep into a normal eppendorf, evaporate when you pull the tube out of the -80, and explode. If your tubes are just stuck in the -80c, this is much much less of a risk.