To overlay two countries, move them so that their centres are over the same point (eg 0, 0). The trick is to make sure that they stay the same size, and don’t get too distorted in the process.
The jamegeo.com country comparison method is:
1. Re-project your country into a UTM zone. This means that you can translate (move) it North or South without enlarging / shrinking the country, or distorting the shape. This only works if your country fits inside one UTM zone… which almost none do.
2. Move the country until the centre sits on the equator. Expect minor change in shape if your country is too fat for its UTM zone.
3. Re-project into a Equirectangular projection (eg WGS84)
4. Move your country along the equator, which you can do in a equirectangular projection without altering the shape.
5. Rescale (a very little bit) to compensate for any shrinking during stage 2.
6. Combine with another country and be very pleased with yourself.