Exactly! If the F-ratio is kept constant, the absolute diameter of aperture will increase when the focal length increases. Therefore, it will let in more light in total, in order to achieve the same illumination per area unit on the sensor plane of the larger sensor (remember that the FOV is kept constant, and hence the total light-gathering area is bigger for the longer focal length). However, the two images will not be equivalent. The image on the larger sensor will have a shorter depth of field. To achieve the same depth of field, the absolute apertures have to be the same, implying a smaller F-number for the larger sensor (remember that the camera-to-subject position as well as FOV are kept constant).
Another approach: imagine having a piece of good old film, say ISO 100. Suppose that your light meter told you to choose F/8, 1/100, you did this, and got a good exposure. Now suppose that you put your film in a camera that exposes half-sized frames on the same roll of film (half-sized by area). Just like you say, you can use the same light meter, and set the camera in the same way, F/8, 1/100. Suppose that you wanted to take a picture from the same spot, with the same field of view. To do this, you would have to choose a lens with a shorter focal length to get the same FOV. With the same F-ratio, the absolute aperture diameter would decrease, just like you say. Since we use the same film in both cases, the total amount of light (number of photons) that it can receive per area unit without overexposing is the same. Lets say that the full-sized frame was correctly exposed and received 1000 billion photons in total. That means that about 500 billion photons exposed the right side, and 500 billion photons exposed the left side (if we assume an evenly lit scene) to give a correct exposure. Now consider the half-sized frame. It is exactly the same size as either side of the full-sized frame. If either side of a full-sized frame is correctly exposed at 500 billion photons, then the half-sized frame must also be correctly exposed by 500 billion photons. This means that at F/8, 1/100 and half the size of the same kind of ISO 100 film, half the amount of photons in total will be used to expose the film, so that the number of photons (amount of light) per area unit is the same (at the same F-number (aperture ratio) the absolute area of the aperture for the half-sized frame will be half of that for the full-sized frame because of the shorter focal length -- it all comes together). This is why we use relative apertures (F numbers), so that the light meter works for all film sizes and focal lengths -- F/8, 1/100 will always yield the same amount of light per area unit, and hence will expose ISO 100 film in the same way, no matter what focal length or film size we choose. However, the images will be different depending on those choices, and in the case of decreasing both the film size and the focal length to receive the same FOV, the depth of field will increase, as will noise because of the smaller amount of photons in total (even though it is the same ISO 100 film in both cases, it will be enlarged twice (area-wise) as much when a half-sized frame was exposed, and this will make noise more visible).
I encourage anyone who does not agree with the above to find a large-sensor and a small sensor camera with lenses that can be chosen for the same FOV on both cameras. Then set both of them to the largest F-number that both lenses allow, the same shutter time and the same ISO. Take the same picture with both cameras, on a scene where there is lots of depth to show out-of-focus areas. Focus on something that is not too far away, to get easy-to-see out-of-focus areas both in front of and behind the focal plane. Then compare the two images. The effect should be very visible, unless you use a very slow aperture.
About the review on the Q: Here is a quote from the review:
"... This relates to the size of the camera sensor which has a crop factor of 5.5x so in full format terms we are talking about "47mm" thus a standard lens. Now this is not the full story. The max. aperture of f/1.9 sound fast and it is terms of light gathering. However, if you apply the cropping factor (f/1.9x5.5 = ~f/10.4) things will appear not all that sexy anymore regarding the depth-of-field potential. ..."
They say exactly what I am trying to explain! Yes, F/1.9 is fast in terms of light gathering, but remember that it describes how much light is gathered per area unit. If the sensor has a smaller area, it will receive less light in total. They even mention that the Q's 8.5 mm F/1.9 lens is equivalent to a 47 mm F/10.4 on FF (meaning that you will get the same FOV and depth of field, and hence the same amount of light in total, meaning less light per area unit and thus a higher ISO for an FF 47 mm F/10.4).
Again, since light meters, F-values and ISO numbers all relate to the amount of light per area unit, it works no matter what format or lens you use. The images will be exposed in the same way if those numbers stay the same, but the images will not be equivalent in terms of depth of field for different frame sizes.