No need to buy specialty labeling products. Here's a cheap, unobtrusive, time-tested solution: write with permanent marker on plain white masking tape on the bottom of the mug, then run through the dishwasher or wash with hot water (yes, the very thing you were afraid would destroy your label). The heat actually turns the adhesive from easy to remove (as masking tape is designed to be) to very hard to remove (by accident at least; it's still easy to remove deliberately).
I discovered this by accident 8 years ago when I labeled some plastic containers with Sharpies and masking tape, then ran them through the dishwasher. It was not at all the easily removable labeling solution I was hoping for. The tape did not wash off, and the Sharpie remains as dark and crisp as the day I labeled it. I still have containers with 8-year-old tape on them to this day. That's when I realized I could lean into this phenomenon and make it a feature, not a bug. I label all kitchen items that need permanent labeling with Sharpies and masking tape in unobtrusive spots, typically the bottom (because let's face it, it's not an attractive addition). In your case, the bottom is an even better place because your coworkers will see your name when the mug is loaded upside down into the dishwasher. In my case, I now have clear labels on my cups and mixing bowls showing their capacity and weight, on my plastic containers showing what lids go with what bases, and so on. I'm a bit of a labeling nerd. I've tested many permanent markers on kitchen items, and they all wash off eventually, but permanent markers plus masking tape makes for a very durable, readable combo. For the record, the tape is nothing special; it was a $1 generic roll from Staples.
And if you do want to remove it? The tape can still be peeled off with minimal effort, but it'll leave a sticky adhesive residue. Not to worry. Brush some oil on it (another life hack: no need to waste good oil; do this with leftover cooking oil after dinner). Let sit a few hours. The adhesive is oil-soluble; it just needs time to dissolve. Then scrub the dissolved mess lightly to loosen it, rinse with hot water, and wash with dish soap and a sponge just like a dirty dish.