In short, we defined our time measurment system that way.

If you leave aside the date change, they’re only 3 hours apart in terms of time of day.

It’s very common for places to adopt the time zone of the nearest human settlement. One island goes to the arctic base to the left, one goes to the right.

Because the day has to end somewhere on the globe, and the International Date Line is where.

Time zones have to start and stop somewhere.

Here's a great Map Men video that talks about time zones, and mentions the Diomede Islands and Kiribati as well:

https://www.youtube.com/watch?v=NBDaLK6EjwI

Nearer the poles you'll have more separation in time zones for the same separation in geographical distance. Closer to the equator you have to travel farther to reach a new time zone.

That's at least my intuitive understanding. Might be wrong...

So these two islands near each other, but also near the pole, will have a lot of separation in different time zones.

We have our time zone "origin" at the prime meridian (Greenwich, UK). As you move one time zone to the east, local time is (generally) an hour later. As you go west, it's an hour earlier. As each time zone spans each direction of the globe, going an ~hour earlier/later along the way, they're eventually going to meet. One direction lost 12 hours, the other gained 12 hours. That's the international date line, where they are 12-(-12)=24 hours apart.

They could have put them in the same time zone (it is a human construct, after all) but since they are associated with two countries, it makes sense to keep each island with its respective country. Since it's right around the opposite side of the prime meridian, it means you're roughly a day apart.

https://googlethatforyou.com?q=Time%20zones

Imagine you're standing ten paces from the south pole. It's 12:00, noon. You then start walking south, and continue in the same direction for twenty paces. You're now on the other side of the world, and it's 00:00, midnight. Twenty paces have a 12 hour time difference.