iPhone SE shuts down in cold weather?

There are operating temperatures to consider. Perhaps you were at or near the acceptable low:

"[…]Use iOS and iPadOS devices where the ambient temperature is between 0º and 35º C (32º to 95º F). Low- or high-temperature conditions might cause your device to change its behavior to regulate its temperature. Using an iOS or iPadOS device in very hot conditions can permanently shorten battery life.

Store your device where the temperature is between -20º and 45º C (-4º to 113º F). Don’t leave your device in your car, because temperatures in parked cars can exceed this range.[…]

Using an iOS or iPadOS device in very cold conditions outside of its operating range might temporarily shorten battery life and could cause your device to turn off. Battery life will return to normal when you bring your device back to higher ambient temperatures.

Your iOS or iPadOS device might not charge or could stop charging if it gets too cold.[…]"

If your iPhone, iPad, or iPod touch gets too hot or too cold - Apple Support

If your iPhone was mostly kept inside your winter clothing, there shouldn’t be a substantial temperature difference.

Per the tech specs:

• Operating ambient temperature: 32° to 95° F (0° to 35° C)
• Nonoperating temperature: −4° to 113° F (−20° to 45° C)

In this case, seems more like the device locked up. That lock-up should not be common.

If you do use an iPhone outside the specified range, there can be problems. Heat can permanently damage the battery. Cold reduces battery capacity. And excessive condensation going from cold to hot and very humid can potentially cause problems.

I’ve very occasionally seen an iPhone or iPad lock up.

Once every several years’ heavy usage, per device, maybe?

Usually after some update, from the few I’ve met.

There are a few “locked-up” reports around here too, which is then a pretty rare occurrence out of a ~billion devices in use.

I don’t mean the following to cause increased concern (as an isolated lockup is probably nothing), but rather as a potential means to alleviate your concern: I usually purchase AppleCare+ coverage (mostly for the accidental damage coverage), as that extends the standard warranty, and increases the coverage.