For food safe I would recommend using Mineral oil and beeswax mixture, pour or dip the mug into mineral oil give it a good coating let it sit and absorb and do this maybe once or twice again until the wood wont absorb any more, adding beeswax after helps but not needed, You can find some beeswax mineral oil mixtures in most woodworking stores, mineral oil on its own does a good job makes the wood look good and is food safe!
On the woodworking forums, this comes up all the time. First, plywood probably isn’t considered safe because of the glue between the plies. Then they’ll tell you not to use any “nut” trees like walnut for people with allergies.
For the finish, cheap mineral oil, not expensive butcher block oil (which is the same thing in different packaging) is by far the best but has it’s price. You have to completely saturate the wood with it and then you have to keep up on it over time. The preferred method is something like flood once an hour for a day, once a day for a week, once a week for a month, once a month for a year, then once a year after that. I may be exaggerating the point, but that’s not too far from the truth.
Good luck! Whenever I think of wooden mugs, it always reminds me of RenFest and bad British accents. Good times…
If you want to use plywood for Food items, then it will probably be best if you Make your own plywood. I do not think any manufactured Plywood is food safe. What Glue did they use to make the Plywood? What types of wood were used to make the plywood? Some manufactures use “recycled” wood from other projects to make their plywood. This gives an end product that has Multiple types of wood and glue and who knows what else was introduced into the mix.
If you want the look of plywood, for safety reasons, construct your own. This way you know what wood and glues are used.
The health of your loved ones could be placed in Jeopardy if using manufactured plywood
Check into the glues used for cutting boards.
Don’t hold me to this, hopefully others will chime in.
Titebond 3 and Some epoxies, you will have to check the labels to verify. West and System 3 are the ones that I have used in the past. Not for this purpose but have had good experiences with them.