I currently write on the pots with paint markers since most of my figs are in pots.
For my trees or grafts, I do the following.
When I initially graft trees, I write the variety on the scion with a garden marker as I prep the scion (label, cut, wrap with parafilm. I don't make labels at the time of the graft anymore because its a lot of work to keep to make all the tags (usually 2 grafts per variety) and to keep them sorted with the scions. Sometimes grafts fail and it's more trash I need to get rid of. It's also faster to have a tray of grafts already prepped scions and I can just go out there and graft without labeling anything.
I will use labels after the grafts succeed. I've used PVC tree labels from amazon but those are junk. They break after 1 year in the sun.
I just recently ordered polyethene tree tags, supposedly they are the softer ones from nurseries that should last years.
I also have the embossed tree tags from amazon that I use for main trees or major grafts that will produce fruit. But these take time to make indoors. I emboss them with a pen at my desk and color them in with a paint marker so they are legible. Its really hard to read when it's only embossed. I'll label both sides and use a longer metal wire than the one included with the tags. I'll put it on the tree and do a figure 8 with the wire so there's extra slack but is taken up, sort of like an electrical service loop. As the branch grows, I don't have to worry about it girdling the tree until it gets really big, the figure 8 should release and give it some slack.
I've considered machine labeling via brother label printers but they seem like more work. I have too many grafts and trees to label all of them using a machine.