At the top of a mountain, say Mt Everest, atmospheric pressure is low. So shouldn't the same thing be true for winter season.
Atmospheric pressure is low at the top of a mountain because the top of a mountain is at a high altitude. A nice first order approximation of the atmosphere is that the atmospheric pressure at some point in the atmosphere is the equal to the weight per unit area of all of the atmosphere above that point. This immediately leads to a roughly exponential decrease in pressure with increased altitude. Extending this first order reduction in pressure with increased altitude to a reduction in pressure with increased latitude is invalid logic.
On to the main question:
Why is air pressure higher in winter than in summer?
This is not the case. Suppose you pick a location in a temperate or polar climate (roughly between 30° and 70° degrees latitude) with a very long history of meteorological records. If you carefully compute the average atmospheric pressure in midwinter versus midsummer, you will likely find that the average pressure is very slightly lower in winter than it is in summer.
What you'll also find is that the extremes of pressure are more likely to occur in winter rather than summer. A nice example of this is wunderground.com's compilation of U.S. city barometric pressure records. All of the highest pressure records occur in winter. However, with the exception of coastal cities that were hit by hurricanes, all of the lowest pressure records also occur in winter.
This suggests that wintertime weather is much more volatile than is summertime weather, at least in the contiguous U.S. This concept applies across locales with temperate or polar climates: In such locales, wintertime weather tends to be much more volatile compared to summertime weather.
There's a marked temperature difference between the Earth's equatorial and polar regions. The Earth's atmosphere works to balance these differences via air flow from the equator to the poles and back. The primary cause of the increased wintertime volatility in temperate and polar locales is that this is when the temperature difference between the equator and the pole is at its greatest and that these are the locales where the conflict is most brought to bear.