Is it possible to stay on Aare in Eichholzwithout being on camping area? And are dogs allowed to come with us on daily trip?
What do you mean by "staying on"?
Dogs must be kept on the leash.
Eichholz camping:
With "stay on" I thought about being there without being on camping area and paying entry.
Dogs on the leash is good.
No, wild camping is not allowed in Switzerland.
Of course, people do it in the mountains, but that's normally very lightweight and just for a night from sunset to sunrise, not camping at a riverside near a formal campsite.
I still don't know what you want to do there. I only know it from walking through the park or observing it from the zoo side of the Aare. In the Summer, especially on weekends and good weather it is full of sunbathers, people who jump in the Aare and float downstream, picnickers, children, BBQers, etc. with tents and campers at the back side. I think it is a normal public park with a restaurant, but how it all works isn't really clear from their website either.
If you want to walk around, sunbath, picnic, etc., it most likely is like any public park, but if you want to know something specific, best to go there and ask in the camping office or restaurant or call the number on their website, link above.
Eichholz is a place to bathe and swim - it is communal ground for all who behave properly :-) There is some parking space.
If you want to use the Eichholz Camping site with caravans, tents and so on, you pay a fee for space and amenities. They do have some cabanas you can rent, too. If caught camping and not paying, you'll get fined.