Are labels associated with a model or a dataset?

I'm not sure if I have this backwards or not, so I'll explain a bit of what is going on.

I want to use Unity's Barracuda api to use an onnx model for classification and detection (depending on the model). Some example projects I've found have a model and labels handy, so it's easy to map the outputs to the labels. ie, if the 5th element has the highest score, i can look up the 5th label and find it is a cat.

However, in my explorations, namely the Onnx Model Zoo as well as downloading and inspecting datasets using Fiftyone, I can only find labels corresponding to a dataset. Perhaps I've answered my own question, but I'll pose it.

When a model is trained, is there a way to find the labels associated with the outputs? Or is that dependant on the dataset it was trained on? (meaning, I should find the dataset it was trained on and use those labels instead)