The small brown marks you find on the inside of new coffee mugs are likely due to a manufacturing process called "crazing." Crazing is the formation of fine cracks in the glaze of the ceramic material used to make the mug. It is a cosmetic issue and does not affect the functionality or safety of the mug.
Crazing can occur when there is a difference in the rate of thermal expansion between the glaze and the ceramic body of the mug. During the manufacturing process, the mug is fired at high temperatures, and as it cools down, the glaze and the ceramic may contract at different rates, causing the glaze to crack slightly.
These marks might be more noticeable in darker-colored mugs because the contrast makes them stand out. Washing the mug with washing up liquid might not remove the marks because they are in the glaze itself and not on the surface.
In terms of safety, the presence of crazing does not make the mug unsafe to use. The cracks are typically very fine and do not compromise the structural integrity of the mug. However, if you're concerned about the appearance, you might consider contacting the manufacturer or the place where you purchased the mug to see if they can provide a replacement.
As long as the mug is washed regularly, it should be perfectly safe to use. Just be aware that the marks are a cosmetic issue and won't go away with regular washing. If you notice any other defects or issues with the mug that might affect its functionality or safety, it would be best to discontinue its use and seek a replacement.