Why do the signature and ciphertext size increase in post-quantum schemes? Why do the signature and ciphertext size increase in post-quantum schemes? And is there any table or comparison as to how much this increase happens for each scheme type (lattice, multivariate, code-based, etc.)
What you say isn't entirely correct. In some post-quantum schemes, the signature or ciphertext is actually smaller than the equivalent-but-classical security level analogue in ECC. See for example the MQ encryption scheme ZHFE (link) or the code-based signature scheme CFS (link). However, in these cases the public key is invariably huge. In other cases we have the opposite: e.g. in the hash-based signature scheme SPHINCS (link), the public key is nice and small, but the signatures are huge. Lattice-based schemes tend to produce public keys of roughly the same size as the ciphertext or signature, but both are noticeably larger than their ECC counterparts. So there is a question there, but it should be rephrased to something like "why is $|\mathsf{pk}| + |\mathsf{s}|$ or $|\mathsf{pk}| + |\mathsf{c}|$ always so large in post-quantum crypto, but not in classical crypto?"

The way all public key schemes work is by relying on hard computational problems with a very definite algebraic structure.* This structure is necessary to guarantee a link between the public and private keys, i.e., only the holder of the private key that matches the public key should be able to decrypt or sign messages. This stands in stark contrast to symmetric crypto, where security relies on breaking -not retaining- algebraic structure and no such link is necessary because there is no distinction between public and private keys. Public key schemes are the subject to more functional constraints than symmetric schemes and as a result there are fewer strategies for their construction. Of those strategies we know to work, only schemes based on number theory and elliptic curves are known to be broken by Shor's quantum algorithm. The others are consequently considered part of post-quantum cryptography. So maybe the best phrasing of your question is something like "why are schemes that are broken by Shor's algorithm, also the schemes with the smallest $|\mathsf{pk}| + |\mathsf{c}|$ and $|\mathsf{pk}| + |\mathsf{s}|$?"

This question has a straightforward answer: both number theory and elliptic curves, or commutative group theory in general, enable a very succinct encoding of computational hardness while retaining a lot of algebraic structure. Unfortunately, this copious algebraic structure comes back to bite you in the ass because it is precisely what allows Shor's algorithm to work.

The good news is that, like Yehuda says, there is no proof that hard algebraic problems that resist attacks on quantum computers cannot have a succinct encoding. So conceivably, it may be possible to design short public key and short signature/ciphertext post-quantum schemes. We just don't know of any that accomplish this.

In terms of a comparison of existing schemes, take a look at eBATS (link), a website that compares operational cost as well as key and message size of various pre- and post-quantum schemes.

*: The exception to this rule is the branch of hash-based signatures. But their notable lack of algebraic structure is also what causes the inherent statefulness (i.e. you have to keep track of all previous signatures to be secure) and what kills the possibility for public key encryption based on hash functions.