Dont have access to Unity/Unreal atm.
@lucas veber (lucky3) : could you upload a screenshot of both Unity and Unreal as a reference?

Sure thing. Actually it's the same issue in Maya in case you're able to test it. Haven't tested with 3dsmax but i'd assume it may be the same.
Here is attached a screenshot.

While praying for this bug not to be forgotten in the dark, i've simplified at maximum the example blend file (attached) to help. It now only contains one mesh and two bones. When exporting to Fbx and importing in other apps, such as Unity, "bone2" is imported with Rot X = 179.8, instead of -90.
Rotating in Edit Mode bone2 1 or 2 degrees in Blender fixes the issue. However it's not a clean solution, it may happen in other situations. Seems to happen when a bone is making 180 degrees x-axis angle with its parent.
After further investigations with the Fbx exporter code, tracking the exported rotations values seems to output correct values. However it's the way these values are read in other app that is the problem. Likely to be an euler issue, if only Fbx used quaternions...

Exported rotations from Blender, using a "print" command in the Fbx script (truncated for readibility):
bone1 [-90, 0, -90]
bone 2 [179.8, 0.115, 0.035]

Imported rotations in Unity (truncated for readibility):
bone1 [0.000, -90, 89.9]
bone 2 [179.8, -0.115, 0.0]

After doing export and import result looks like this:

So this issue may have been replaced by another issue

I was reading this ancient post about FBX import/export in Blender, and it mentions "FBX bones are not aligned along the same axis as Blender bones (former are along -X axis I think, while later are along +Y axis)".
If this was never fixed, it might be what's causing the weirdness displayed in the previous sreenshot