Let's say here we have two sensors, one bigger and one smaller, both of them have same pixel size and everything is identical except the sensor size. And now we are photographing the same object with the two sensors using identical camera lens. Bigger sensor has larger FOV compared to the smaller one and the total light collected will be greater. Then, we crop the image captured by the bigger sensor to match the image captured by the smaller sensor. Will the SNR for both image be the same? Because a lot of sources on the Internet stated that larger sensor is better because it will capture more signal, resulting in less noise. But I don't see other advantages here except the FOV part, in which larger sensor has to spend less time on capturing a particular object that the smaller sensor may have to use mosaic method on. https://www.youtube.com/watch?v=WNoO6qJS7LM The concept is explained here at 10:53 but I don't really get it.
Then, we crop the image captured by the bigger sensor to match the image captured by the smaller sensor. Will the SNR for both image be the same?
If you crop the image from the larger sensor to only the portion that was captured by the smaller sensor, you've given away the "more signal" captured by the larger sensor. Assuming the same ISO, aperture, and exposure time both sensors collected the same amount of light per unit area. In other words each sensor captured the same amount of light per sensel (a/k/a photosites or pixel wells). The reason the larger sensor captured more light is because it has more pixels spread over a larger area and those pixels captured more light from a greater angle of view.
Because a lot of sources on the Internet stated that larger sensor is better because it will capture more signal, resulting in less noise.
Assuming both sensors use the same technology, materials, etc. and have identical pixel pitch then at the pixel level there is no S/N ratio advantage for the larger sensor. The reason the complete image from the larger sensor will still look less noisy is because it will not be enlarged as much as the image from the smaller sensor in order for both images to be displayed at the same size. The viewer will be less able to discriminate individual pixels, so noise will be averaged out by the surrounding pixels that the viewer all sees blended together as a single point. But if you crop the larger image to match the smaller one, both will be enlarged by the same factor to get to the same display size. So again, there will be no advantage for the larger sensor.
Given your assumption that both sensors have the same sensel size, a lot of what you read and hear on the internet about the advantage of larger sensors isn't as much the case as when a larger sensor also has lower pixel density/larger pixel wells.
Let's take your theoretical pair of sensors and turn it on its head. Let's say both sensors have the same number of photosites. That means if the larger sensor is full frame and the smaller sensor is 1.5X APS-C, then the area of each sensel on the FF sensor is 2.25X larger than the area of each sensel on the APS-C sensor. Again, assuming both sensors use identical materials, technology, etc. then the sensels of the larger FF sensor would have a full well capacity 2.25X greater than the full well capacity of the smaller sensels on the APS-C sensor.
This allows each sensel to collect more light at the same exposure parameters (ISO, Av, Tv) without blowing the highlights as soon as the smaller sensels with lower full well capacity would. In other words, all other things being equal, sensors with larger photosites will have greater dynamic range. If exposed so that the higher full well capacity is utilized, this higher full well capacity combined with the same amount of read noise and less "shot noise" (caused by the random Poisson distribution of photons in a stream of light) means a higher signal-to-noise (S/N) ratio.
Because the pixel density of larger sensors is smaller (the "sensels" are bigger, so each sensel captures more light)... If your small sensor was just a cut-out of a large sensor the 12Mpx of a smartphone camera sensor would correspond to a full-frame sensor of several hundred megapixels.
The area ratio between a full-frame and an APS-C sensor is 1.6²=2.56, so a full-frame sensor with the same pixel count as an APS-C sensor has sensels that are 2.56 times as big and catch 2.56 times more light. In practice the full-frame sensors often have a bit more pixels than APS-C sensors of the same generation, but not 2.56 times more... The top-of-line APS-C sensors are around 32Mpix, at the same sensel density the full-frame sensors (1.6²=2.5 times more area) would be 82Mpix sensors (the biggest I can find is 61Mpix).
As you ask that question, the results will be identical at the same aperture opening of the identical lens.
So where does the difference come in? For the crop sensor, that lens is overkill. You have far too much glass designed to illuminate a much too large image circle. Crop sensors just don't have the same pixel densities as larger sensors: for a crop factor change of 3 or 4, you usually see an MP change of at most 2 or so (like 40MP vs 20MP), not of 9 or 16. So in spite of getting 1/9 or 1/16 of light (at same aperture number), you don't get that many fewer pixels.
The ugly truth in the end is that large sensors do not capture more light. Large sensors have a larger image circle, a larger image circle requires a larger lens, and the larger lens then captures more light. Of course you can use an oversized lens also with a crop sensor but then you carry around a lot of unnecessary weight. You can utilise some of that weight with a "speed booster" that concentrates the bigger image circle on a smaller one, giving a wider angle and an effectively lower aperture number for the new wider effective focal length.