
Is the pottery mug you purchased dishwasher safe?
There is pottery that is dishwasher safe. If I don't say otherwise, yes, my pottery is dishwasher safe. All my wares are fired to what's called vitrification. That really just means that the piece of pottery was fired to a temperature that makes it no longer hold the water and split from water absorption.
I have two friends that buy mugs from me. One always handwashes her pieces. The other only ever uses the dishwasher. The one who uses the dishwasher has less chipping and breakage than the friend that handwashes her pieces. There are pieces that do chip and break. One does need to be careful, but I know that even if you are careful we still do misjudge things. We don't mean to be careless, and we aren't careless. Life just happens.
When there is gold an a piece, I can't promise that the piece won't wear over time. Gold is soft and is fired in a separate firing. I would recommend that you handwash pieces with the bright gold. (It is real gold on my pieces, btw.)
What do I recommend? I don't know. I do want your pieces to remain whole. Do what you feel comfortable with. I always handwash my mugs, but that's because that's my only option.
Note: I can't speak for pottery from elsewhere. You will need to use your own discretion.