In predicting the outcome of a future match, we are considering our random variable to be intrinsically multivariate, the pointwise likelihood is then p(\mathbf{y_i}|x_i, \theta) where \mathbf{y_i} = (y_i^{(h)}, y_i^{(a)}), \theta are the model parameters and x_i would encode which team plays home and which team corresponds to each superindex.
It is therefore different from the aggregate outcome as the aggregate outcome would not distinguish between home and away goals. That is, if we only look at the aggregate, \mathbf{y_i}=(1,2) is equivalent to \mathbf{y_i}=(2,1), \mathbf{y_i}=(3,0) and \mathbf{y_i}=(0,3). I tried to be more specific with the pointwise log likelihoods that correspond to these two cases which gets more mathematical, I hope it helps and the notation is not too obscure.
In predicting one observation (which now I realize is probably bad naming), the pointwise likelihood is p(y_i|x_i, \theta) now with y_i being a scalar and x_i and \theta encoding the same info.
The specifics of writing the exact pointwise likelihood function would still depend on the parametrization, but to define the predictive task, the definition of y_i is what matters. One possible parametrization for the match case would be:
and for the observation case:
in both cases, atts and defs are vectors of length team containing the team performaces and \mathbf{t_i} contains the team indexes (needed to get the correct values from atts and defs). In this second case, h_i would be a binary variable being 1 if we aim to predict the goals scored by the home team and 0 if we aim to predict the goals scored by the away team. Therefore h_i determines whether or not to include the home effect and it also determines which component of \mathbf{t_i} is to be used to index atts and which should index defs.