Is it ok that a suspension seat post like the one shown below (source: cyclande.com) is actually a bit loose (moves a bit left-right and probably also up-down).
The vendor told that for such seats it's OK, but I'm not totally sure.
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An up-down movement is perfectly fine – that's what suspension seat posts are designed for!
For the side movement it's a bit more complicated. A suspension seat post is basically two tubes sliding into each other with some suspension mechanism. This requires two things: the tubes need a little clearance to move within each other and they need a mechanism that prevents the tubes from rotating towards each other – let's call it anti-rotation machanism. The latter can be for example something like a bolt sliding in a notch or tubes with a non-circular cross-section.
The bad thing is, those two elements are somewhat conflicting. An effective anti-rotation mechanism requires the least possible clearance, which may inhibit the original purpose of the suspension seat post, which is to allow for up-down motion of the tubes. Therefore, the manufacturers have to find a trade-off between suspension performance and precision. Such precision is expensive and would quite surely also make the seat post heavier. Especially the products from the lower end of the price range might quite surely sacrifice some of the precision for a more "reliable" suspension performance.
So to answer the question: yes, some side rotation or wiggle (something like +-2-3 degrees) should be quite common.
I have used a couple of these seats over the years - the ones I had did have some movement - noticeable when riding. Although not ideal, you do get used to it. It comes down to cost - it would be more expensive to precision manufacture them with no movement, but at some point people would stop buying, and another point, might as well buy a full suspension bike.
Yes, it's ok and to be expected. Even expensive dropper posts, which work via the same basic principles as telescoping suspension seatposts, have a bit of side to side movement. For the suspension seatposts, just ensure that the collar at the top of the stationary (outer) portion of the seatpost is firmly hand tight.