In Photoshop the add blending mode is named LinearDodge(Add). The reason why it might look to do incorrect results is due to the color space it operates in. To handle the data we need to work in linear space. If you want to do non-clamped, fully linear compositing, you need to use 32bit-mode, at all the times.
Export your layers as MultilayerEXR, import into Photoshop with EXR-IO addon. Photoshop will work in linear space when set to 32bit depth, which it is by default when opening EXR.
Set your layers to multiply and add like this:
You get the correct result. Next you want to convert this into Smart object, then convert to 8/16bits without merging layers and do whatever you want with it. Like this you can still work inside the Smart object in linear space.
You can also work in linear space when using 8/16bit depth, but you need to set it inside Edit > ColorSettings, because Photoshop does not default to it in 8/16bit depth. Use Custom profile with Gamma 1.0. Your monitor probably displays in sRGB, so you need to check View > Proof Colors, to see it converted into your monitors color space. Note that some color spaces do not offer the same gamut, so there will be losses when converting, which is un-avoidable. sRGB and linearRGB have the same color gamut, only the gamma curve is different, so the color space is no problem.
Photoshop is designed to handle whatever color space (CMYK, AdobeRGB, Linear, etc.) and does not only work in display-referred space.
But only in 32bit depth it is unclamped. Working in 8/16bit will result always in clamping and incorrect result because of this, not because of color space.
Explanation why this does not work in 8/16 bits:
When exporting the layers as png or tiff in 8 or 16bits, you loose accuracy there. You can export them as sRGB and composite in Photoshop in sRGB space with no problems. But color blending operations will result in clamping when not in 32bit, so another loss of accuracy.
You might loose some sharp highlights in 16bits and probably your image will look very different when using 8bits.