PCA is actually just a rotation. Seriously, that's all: it's a clever way to spin the data around onto a new basis. This basis has properties that make it useful as a pre-processing step for several procedures.
The basis is orthonormal. This is incredibly useful if your features exhibit multicolinearity (two or more features are linearly dependent): applying PCA is guaranteed to give you a basis where this is no longer a problem. This procedure is known as principal component regression
The basis vectors are meaningful with respect to the spread of the data: they are the eigenvectors of the covariance matrix. This second property gives rise to PCA's famous utility as a dimensionality reduction technique: after rotating the data, projecting the data onto a subset of the basis vectors associated with a significant portion of the total variance yields a lower dimensional representation that (often) retains (most of) the (interesting) structural properties of the data.
So: is it a learning algorithm? This is sort of a philosophical question. What makes something a learning algorithm? Certainly PCA isn't a "supervised" learning algorithm since we can do it with or without a target variable, and we generally associate "unsupervised" techniques with clustering.
Yes, PCA is a preprocessing procedure. But before you write it off completely as not "learning" something, I'd like you to consider the following: PCA can be calculated by literally taking the eigenvectors of the covariance matrix, but this is not how it's generally done in practice. A numerically equivalent and more computationally efficient procedure is to just take the SVD of the data. Therefore, PCA is just a specific application of SVD, so asking if PCA is a learning algorithm is really asking if SVD is a learning algorithm.
Now, although you may feel comfortable writing off PCA as not a learning algorithm, here's why you should be less comfortable doing the same with SVD: it is a surprisingly powerful method for topic modeling and collaborative filtering. The properties of SVD that make it useful for these applications are exactly the same properties that make it useful for dimensionality reduction (i.e. PCA).
SVD is a generalization of the eigendecomposition, and that too is extremely powerful even as a constrained version of SVD. You can perform community detection on a graph by looking at the eigenvectors of the adjacency matrix, or determine the steady-state probabilities of a markov model by looking at the eigenvectors of the transition matrix, which coincidentally is also essentially how PageRank is calculated.
Under the hood, PCA is performing a simple linear algebra operation. But, this is exactly the same operation that underlies a lot of applications that most people wouldn't question applying the label "machine learning" to. This class of algorithms is called Matrix Factorization, and even extends to sophisticated techniques like word2vec: indeed, you can actually get word2vec-like results by literally just applying PCA to a word co-occrrence matrix. Generalizing again, another word for the results of PCA is an embedding. Word2vec is probably the most famous example of an embedding, but constructing embeddings (as intermediaries) is also an important component of the encoder-decoder architecture used in RNNs and GANs, which are the bleeding edge of ML research right now.
So back to your question: is PCA a "machine learning algorithm?" Well, if it's not, you should be prepared to say the same about collaborative filtering, topic modeling, community detection, network centrality, and embedding models as well.
Just because it's simple linear algebra doesn't mean it isn't also magic.