Do the standard deviation and mean use the same units?

The standard deviation and variance have the same units or are both unitless, the same unit used for the observations within the dataset. If you were calculating the standard deviation of a dataset measuring the weight of apples, you would express the standard deviation and variance in terms of g, unless you had extremely heavy apples then kg.

Yes, they have the same unit of measure.

Data are normally present as mean ± standard deviation of the same quantity. For example, a group of 100 students could be aged 13 ± 5 years, where 13 years is the mean age, and 5 years is the standard deviation of the students' ages.