I think you are seeing the difference in the system limits between a 20MP camera with an APS-C (23.2 X 15.4 mm) sensor that has three times the areal size of a 20MP camera with a 1" type (13.2 x 8.8 mm) sensor. You're also likely seeing the difference between a higher quality lens and a more budget friendly lens.

The background in the image from the Panasonic LX-10 is not necessarily "out of focus", it just isn't as sharp as the background in the shot from the Sony α5000.

Since both cameras have the same number of pixels, the Panasonic has photosites (a/k/a sensels or pixel wells) that are considerably smaller than the Sony. The Sony's pixel pitch is 1.75X larger than the Panasonic's. This affects the final image in several ways.

• Due to the smaller size of the photosites on the smaller camera, the diffraction limited aperture is encountered at a lower f-number. If you use the same f-number with both cameras, the LX-10 will suffer more from diffraction than the α5000 will.
• Due to the greater enlargement ratio needed to view images from the smaller sensor camera at the same size as images from the larger sensor, a lens used on the smaller sensor would have to be able to resolve 1.75X more linear detail than the lens used on the larger sensor just to get the same overall amount of "sharpness". This is because any blur in the images from the smaller sensor is being enlarged 1.75X more than the blur in the images from the larger sensor.
• Although we don't know what specific lens you used with your Sony α5000, pretty much any of Sony's e-mount lenses would likely be higher in optical quality than the LX-10's permanently attached 10.25-30.75mm f/1.4-2.8 zoom lens. It's not a "bad" or "terrible" lens, but it isn't as good as what one would typically put on the front of modern interchangeable lens cameras. Due to the camera's approximate 2.34X "crop factor", this gives a lens "equivalent" to how a 24-72mm f/3.3-6.6 would act on a full frame camera. Of course your α5000 was also a crop body, but it only has a 1.55X crop factor.

All of these factors interact to limit a camera with a smaller sensor and smaller pixels and a lesser lens to lower maximum sharpness than a camera with a larger sensor, larger pixels, and a better lens can achieve.

In the specific cases of your two example photos, it seems the LX-10 shot is focused a little bit in front of you, while the α5000 shot is focused just a tad behind you. It also looks like atmospheric and surface conditions (temperature variations that cause atmospheric "waviness" during warmer weather, haze from moisture and particulate matter in the air, and although it is hard to tell for certain, the distant peaks in the α5000 shot appear to be bare rock while the peak in the LX-10 shot is covered in trees) might have also affected the comparative results.