It’s definitely travelling through the ground. The problem is the structure of the house is amplifying it like a drum (through the walls and ceiling)It's a tricky one but if you're feeling the sound of kids running then it must be travelling through the ground. It could be something you can't change like concrete foundations joining the properties (why I wouldn't know). Or maybe it's because the paths running down the side of the fence are physically coupled by the fence posts and the foundation. Separating these will reduce the physical transmission....certainly needs more thought into the correct solution but that's the likely culprit for what you've described.
Soundproofing sounds difficult and this is what the 'specialist' companies want to sell their services on but a lot of it is just common sense when you think about how sound travels. Soundproofing is just looking for weaknesses and addressing those. Some of it is easier to DIY than others. Like in an old house make sure every gap is filled with caulk, or thicken up the walls by using two layers of plasterboard (stuck together with carpet adhesive) which adds mass and vibrational dampening.
We did consider paying for an acoustic survey of the property, but again it’s just extra money that we don’t want to spend.