In theory, you should be able to run a supervised classification on many mosaicked tiles based on a limited area's training samples. Whichever algorithm you use (max likelihood, isocluster, etc.) will iterate through the merged bands and identify pixels with similar spectral characteristics to the ones you sampled for the training sites.
In practice, you'll typically want to sample from across the entire study area - meaning you'll probably want at least a majority of the tiles represented for each class. The reason is that this will reduce variation that may occur as a result in different factors for each set of tiles, such as distortion, radiance, and atmospheric effects. For such a large study area, an unsupervised classification might be easier, depending on how much detail/accuracy you need or what you're studying, but with so many tiles you would likely get a lot of noise in the output if you weren't careful in pre-processing. I would suggest taking more training data and running a supervised class.