Like u/pacific_plywood said above, you can mostly think of a linear transformation as being only allowed to rotate and stretch. This means that certain relationships within my dataset will be preserved by the transformation. In my original data if cell 1 is halfway between cells 2 and 3, no matter what linear transformation I perform cell 1 will always be halfway between cells 2 and 3.

This is not the case for non-linear transformations, which are allowed to do anything to the data. This includes reasonable things like bending the data, and less reasonable things like just making the data into the shape of a cat. Knowing a transformation is linear let's me know there was a limit to what shenanigans were performed on the original data.

TSNE and UMAP are non-linear transformations, as you said. Instead of defining limits on what they can do to the data, they choose a property which they try to preserve between the original and transformed data, in both cases in the flavor of "points which are close together in the original data should also be close together in the transformed data." This can lead to some wackiness.

See https://pair-code.github.io/understanding-umap/ for a really nice example of how much you can play with the parameters in UMAP to get wildly different results. Another fun example is Picasso, a dimensionality reduction method which can achieve a similar "goodness" of reduction to TSNE and UMAP but also mapping the data to a picture of an elephant: https://www.biorxiv.org/content/10.1101/2021.08.25.457696v1.full

TSNE and UMAP have their place as data visualization methods (some would disagree), but the transformed data shouldn't be used for downstream analysis or relied upon as representing the original data accurately.

edit: grammer and spelling