Why the Shapes Contours Colors are not the same than the shape himself when I export them as PNG

Try if your can export your layered vector content as a layered PSD with Art Optimized for anti-aliasing and a resolution of 72 ppi or multiples thereoff (144, 288).

I guess it should be possible to batch open the png files in Photoshop and expand each objects shape (increase opacity of partially opaque pixels) in printing this would be similar to trapping.

In illustrator add a thin stroke would make objects slightly larger, and touch on overlapping rather than the hairline spaces you have. Snapping artwork to pixels would be a better way to go but diagonals and arcs may still be a problem. 

Are you exporting each shape tto PNG and then having them layer? Can you not make all the objects to a PNG?

This is antialiasing. You need to work pixel perfect by aligning to pixel grid which will help with vertical strokes, but curves and diagonals may get mathematical floating point operations that cause visual hairlines. (The problem is 50% of 50% is not zero, so it's a bit of a mathematical problem.)
PNG also is not true transparency only alpha transparency of edge pixels and this has effect on how the edges blend. 
You can rasterise the image from illustrator, before exporting it, or have the one shape under the smaller shape rather than having your shapes meet edge to edge.

If you can explain more why you need each layer as a seperate PNG maybe we can help you think of a workflow that works for you.

If you are trying to combine images in Illustrator, I would recommend Photoshop for that task.

Please show us the objects in the layers panel and select each of the affected objects and then post a screenshot of object and appearance panel.

Hugo,

I believe it can be an anti aliasing issue, possibly caused by PNG images that are not contained within whole pixels, owing to positions and/or sizes with decimals.

The following may sound unforgivably outdated and boring, sorry.

If you wish to have PNGs (PNG24 (also (little) known as PNG32, it holds 24 bit colour and 8 bit Alpha channel (transparency)), of course) look crisp and clean, at least when it is (also) to be used at moderate screen resolutions, it is important to have the images in the exact desired final pixel x pixel size, or at sizes that are powers of 2 times as large (2x, 4x, 8x, and so on, the larger values can improve the appearance on high resolution screens and still ensure best possible appearance at low resolution screens); forget about image resolution which may actually lead to wrong sizes and hence blurriness, or work at 72PPI or powers of 2 times as large (144PPI, 266PPI, 576PPI, and so on).

It is easiest and safest to work at the desired size when creating the artwork.

It is important to remember that a raster image represents the whole appearance, including strokes, so to make sure you get it right you can click Show Preview Bounds in the General Preferences (and untick it afterwards).

A very common unsuitable way is to Export to PNG (remember to use PNG24 and use Transparency for artwork to be in front of different backgrounds) with a medium or high resolution, such as 300PPI.

And a common misunderstanding: (almost) 11 out of 10 times, a statement like "I created the document at 300 PPI" means that the value is chosen in Effect>Document Raster Effects Settings; however that only means that the (current) resolution of any raster effects applied to the vector artwork, such as (any kind of) Blur, is set to that value (and only unless/until the value is changed to something else); when zooming in, this resolution can be seen in contrast to and on the background of the vector artwork. So this setting has nothing whatsoever to do with the actual resolution of a raster image created from the (vector) artwork (but it ought to (at least) match it).

For clean and crisp artwork avoid JPEG; to get the colours right, you may wish/need to finalize by opening the PNG in PS (Photoshop) and attach the (missing) colour profile (such as sRGB) there.

It is also important to have the artwork and also the Artboard(s) placed fully within integer/whole pixel X and Y values in the Workspace, which means that the X and Y values at the corners must be integer; this can be ensured by using one of the corner Reference Points in the Transform palette, and then checking that all the values X, Y, W, and H, are integer (the centre Reference Point can only be used if both W and H are even numbers).

If you have issues with the Artboard(s), make sure the rulers are Global Rulers (RightClick), if not change from Artboard to Global Rulers; and to align your artboards, click the Rearrange All Artboards button in the Artboards panel. First posted by Ton here,
https://community.adobe.com/t5/illustrator-discussions/why-is-illustrator-not-exporting-artboard-size/m-p/13350220#M343886

Otherwise the resulting image will become a bit wider/taller and the extension(s) will be empty and therefore be (partially) transparent/white.

Therefore, a safe way is to create the artwork at the final pixel x pixel size and use a corresponding Artboard, then use the Legacity Save for Web (where you can look in the Image Size window for size confirmation and possibly multiply by 2, 4, 8, whatever), or use Export at 72PPI (or 144/288/576/whatever PPI), or use Export for Screens (in either way). In either case, use the relevant optimization (available with both ways); it is also convenient to have 72PPI (or 144/288/576/whatever PPI) in the Effect>Document Raster Effect Settings.

If you have pure vector artwork, you can relax a bit and have the artwork/Artboard at any size (the Artboard must have the same proportions as the final image), then use the Legacity Save for Web and set either Width or Height in the Image Size and Apply (make sure the other value is also correct).

The Legacy Save for Web may be an old carthorse, but it knows its way home, even if the driver is drunk and sleeping it off in the hay in the back.