The other things posted here are true, but the big reason is that winter doesn’t last long enough for us to meaningfully deplete the oxygen in the air. If you look at carbon dioxide, of which there is much less, you do indeed see a [very noticeable seasonal cycle](https://en.wikipedia.org/wiki/Keeling_Curve#/media/File:Mauna_Loa_CO2_monthly_mean_concentration.svg) where CO2 levels peak in the winter of the northern hemisphere (where most land is, and thus most land plants live).
This seasonality happens worldwide, though, because the atmosphere effectively mixes gases from every part of the world to every other part over timescales of weeks to months.
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