judging by the middle image you're working with RGB images and export to CMYK – right?
If your output is intented to be printed, then there's not much you can do, since CMYK has a smaller gamut, meaning you can't replicate every RGB color in CMYK (imagine a bright 255-Red – you can't print that with CMYK).
The middle black images has another issue – if it's an RGB image then it gets converted to 4c black, so for example resulting in CMYK 83 / 67 / 51 / 95. Your background solid color seems to be InDesign's [Black], which is 0 / 0 / 0 / 100. So there you have your difference.
In that szenario, either extend your image in Photoshop (safer version) or make sure the colors match after export. The values depend on what profiles you're using in InDesign – you also could set up a RGB 0 / 0 / 0 color swatch in InDesign, which then gets converted on export – but then the RGB profiles (image vs. InDesign doc) have to match to get the same CMYK values.
If it's for web only, change your transparency blending space to RGB and export without color conversions to CMYK.
The color in general doesn’t seem to be outside of the printable gamut, so it seems more like mismatched color settings. Try exporting as PDF/X-4, which will include an output intent for the preview in Acrobat. Also check your InDesign Color Settings, there is a bug in the latest CC updates that turns off color management.