The refresh rate of the monitor is not relevant for the recording. Relevant for the recording is the fps of the source you're recording.
However, there is a relationship between the fps of the monitor and the fps a game is generating, so there is an indirect relationship between the refresh rate of the monitor and the recording. The relationship between the fps of the monitor and the fps of the game is vsync. Vsync is a feature to avoid screen tearing on the monitor.
Screen tearing happens if the refresh rate of the monitor is not equal to the fps of the game. If the fps and refresh rate doesn't match, it happens the game generated a new frame while the previous frame is still being transmitted to the LEDs of the monitor. In this case, the upper part of the currently displayed image on the screen is from the previous frame, and the lower part is from the new frame. Since they are different, you see a horizontal seam between the upper and lower part, which is called screen tearing. Screen tearing is visually unpleasant, and I might say for me, it provokes the same symptoms as motion sickness for me personally.
To avoid screen tearing, the game can be told to synchronize itself to the video data transfer, which is called vsync. If it generated a frame, it doesn't transmit immediately to the monitor. Instead, it waits until the previous frame has been transmitted completely and only after that it releases the new frame. This short delay lowers the fps a bit, because the game has to wait with rendering, but it avoids the display of teared images.
Now, if you capture a game with OBS' game capture, you're not grabbing the (possibly teared) data back from the monitor. Instead, you grab the data directly from the frame buffer of the game. It's the frame buffer that is also being sent to the monitor. It has its own synchronized access method, so as far as I know, there is no vsync or screen tearing issue with game capture. That means, if you turn off vsync to achieve higher overall fps of your game, you get screen tearing on your monitor, but not in your recording. Vsync is to sync the game to the monitor, not to capturing.
There is no difference between game capture and display capture. Screen tearing takes place, because a displayed frame may change while the data is being transmitted. Within the frame buffers of the computer, this data transfer is made instantly, so there is no time for a frame to change in between. Of course copying video data isn't really instantly, but there are internal methods to make it look instantly.
There is a method to avoid the fps drop due to vsync: Activate triple buffering in the graphics driver. At least the Nvidia driver has this function. It activates a third frame buffer that is used for rendering while the GPU has to wait for vsync otherwise, so no rendering time is lost.
