Predicting Multiple Values Values Using Time Series Forecasting
Thanks for this question I think it is a nice use case to play with time series forecasting in all (or many) of its types. 
As you suggest, there are several possible approaches, and all of them are valid hypothesis a priori to check and validate with your goal. Answering the question:

yes, it is possible to build a single model to predict the sales amount of your products at once (aka a multi-output forecasting model) where each of your products would be a feature to predict --&gt; https://www.tensorflow.org/tutorials/structured_data/time_series#multi-output_models, and you can whether predict a single time-step or a multi time-step forecasting

I would say building a model for each product makes sense, as you will potentially have many more customers than products throughout of time; this way, for each product you will have several customers who bought it, and modelling the clients characteristics for a certain type of product sounds reasonable; in this case, you have a single-feature to predict (in the same source of info you have this model type, agsain for both single time-step or multi time-step forecasting: https://www.tensorflow.org/tutorials/structured_data/time_series#single_step_models