Which machine learning problem is this?

Technically this is sequence labeling, the most common application being Named Entity Recognition.

However it looks like in this case you're trying to solve a problem of coreference resolution, which is a quite difficult task in general. I think this usually involves a more complex model than simple sequence labeling, but I'm not an expert in this. You might want to search around this topic, there are certainly some relevant papers and tools about it.

There are several different ways to frame the problem.

One way is multiclass classification. The goal would be to assign a single discrete label to every phrase. In order to get phrases, you'll have to build a parse tree first. You did not list of all of the labels but let's assume they are all nouns. Then you'll need a Part-Of-Speech Tagger (POS Tagger) to find all noun phrases. Each noun phrase can be predicted to belong a class.

@Erwan probably gave you a better idea, but for a simpler problem, when each token has a separate class, this can be viewed as a Part of Speech Tagging, which is essentially a per-token multiclass problem. This assumes each token can be mapped only to one correct class. In such case the LSTM's output would be size, e.g. (1, seq_length, hidden_dim), which you reshape to (seq_length, hidden_dim), and seq_length becomes the batch dimension. You feed this output into a linear layers, to get a batch of outputs size seq_length. Now you can use NegativeLogLikelihood loss function for this batch. For each token you will have a one-hot vector (multiclass problem) of labels.