So, I take a RAW photo and save it as a psd file with Photoshop.
Then in lightroom I open the same RAW photo and the psd file too and then apply the same filter to both photos.
The result is that both photos look different when I apply the filter. The tones change.
Can someone help me? I already researched and try matching the color settings between photoshop and lightroom and can't seem to figure it out.
A raw file and a rendered RGB file (psd) are two very different animals. They will react differently to further adjustments - the same adjustment will produce very different results. This is normal and expected and unavoidable.
Any numerical adjustment is relative to the color space the file is encoded in. Even two RGB files in different color spaces will show this - you can try this for yourself in Photoshop.