Easiest way is probably to double your world's orbital length around its sun.
Our seasons happen because our planet is tilted in respect to its orbital plane (the imaginary plane we would get if we made a flat plane that included all points of the Earth's orbit). The planet's orbit causes that tilt to be in a different direction in respect to the sun as it moves around the sun. The tilt's direction isn't actually changing, but this orbital motion and the tilt mean that at different parts of the year, a different line around the planet is what will be closest to the Sun.
So, double the planet's orbital period. You'll still have the same seasons, but lasting twice as long. Of course, this now means that the orbital period is two earth-years longs, but that should be fine. Just have your civilizations count their years starting on each solstice, irrespective of whether it's the summer or the winter solstice.
The major issue with this is that your sun will have to be hotter, since the increased orbital period will also mean that the planet is farther away, but that shouldn't be a problem as long as you aren't diving into the science behind this planet's orbit. Even some light handwaving should give the desired effect of having the same climate, and the most attention you'd ever need to bring to it is that the two "years" don't have the same stars in the sky.
If you're going for a realistic star for this setting, you could ask the question of what kind of star would have a goldilocks zone that's twice the orbital period of our own.
For the distance, you will want to take the Earth's distance from its orbital point and multiply that distance by the square root of two (very roughly rounds to 1.4). That should give you the radius you need to double the orbit's circumference (and thus its orbital period). This will give you a very rough approximation because I'm ignoring the possibility of an elliptical orbit and the gravitations effects of other bodies, but it should get you in the right neighborhood at least.