Yes. Stacking multiple exposures of the same scene is a valid way to reduce noise, and it’s commonly used in astrophotography.

For pure noise reduction, averaging aligned frames is generally the most effective way to improve signal-to-noise ratio. Median stacking is useful when you need a simple way to reject outliers, but it usually won’t give the best final SNR because it picks a single middle value rather than fully using all the data.

A better approach is typically mean/average stacking with some form of outlier rejection, such as sigma rejection. That lets software average the consistent signal while discarding or down-weighting rogue pixel values and other anomalies.

If you have a good statistical model of the noise, you may be able to do slightly better still, but in practice the biggest gains usually come from capturing more frames.

The small frame-to-frame shifts you noticed can also be useful. When exploited deliberately, this falls under super-resolution. That can improve apparent detail or sampling if the software is designed for it.

So: yes, software exists for this, especially in astrophotography, and the key terms to look for are image stacking, sigma rejection, and super-resolution.