Does MonoGame not allow large textures like (20,000 x 1,000)?

If you are not using mipmaps or texture compression then power of 2 resolution it is not that important.

Furthermore resolution is directX feature level limitation. Dx10 max texture width/length is 8k. For more details refer to https://msdn.microsoft.com/en-us/library/windows/desktop/ff476876(v=vs.85).aspx At DX11 you will be fine up to 16kx16k. If that isn´t enough then there are ways how to use multiple texture properly mapped to one UV set.