The sun’s position changes throughout the year, but that is the vertical position (altitude from the horizon). One every single day of the year at noon, the sun will be exactly dead north or south depending on where you are in the and what day it is. In summer, the sun might be dead south of you, and high in the sky. In winter however, the sun will still be dead south, but low in the sky. It is only the vertical position of the sun that changes day to day.
This means that a sundial will stay accurate, but the vertical position does not matter. Sundials track the horizontal movement of the sun. The sun will cast longer and shorter shadows throughout the year, but the shadow will always be at the same horizontal angle.
Now, /u/Skusci makes a good point about the Equation of Time, which does offset the times by give or take 14-16 minutes throughout the year. However, that is due to a different phenomenon than the one you seem to be addressing.
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