Does Bitcoin Core validate signatures by default? There have been claims recently of Bitcoin Core not validating the signatures in a block by default. If this is true, can Bitcoin Core even claim to be a full node?
Bitcoin Core does validate all signatures after a certain point, and can be configured to validate all signatures in history if so desired.
The assumevalid feature only disables signature validation for blocks prior to a specific configured block which is updated for each version. This block is typically several (tens of) thousand blocks deep by the time the release is actually made. Any block which is an ancestor of that particular block (and that block is in the main chain) will not have the signatures checked. All blocks that come after it will have their signatures checked. This means that once your node is synced, all new blocks will have the signatures checked.
Furthermore, this assumevalid is not a checkpoint nor does it operate on block heights. If, for some reason, there was a reorg which removed the assumevalid block from the main chain, then all signatures in the blockchain would be validated. It is merely an optimization and does not affect the security of your node.

can Bitcoin core even claim to be a full node?

Yes, there is more to being a full node than validating signatures. There is more to validation than just signatures. Bitcoin Core still checks the PoW, that the outputs being spent existed, etc.