Does the chain become computationally more expensive to validate over time? Since there is only one chain per coin, doesn't that mean it becomes computationally more expensive to validate the chain the older the coin is and the longer the chain becomes?
Validation of proof of work is extremely cheap, and scales linearly with the length of the chain. That is to say, validating the proof of work for 2N blocks will take only twice as long as validating the proof of work for N blocks.

The part that does not scale linearly over the life of the chain is the transactions themselves - signature validation is expensive, and as networks get more busy, validating the transaction signatures and the utxo set takes up the bulk of the block validation time. These are limited by the SIGOPS limit of a block, which limits how many signature verification operations a block may have
(although it is not an exact 1:1 correlation, but more of an upper limit/yardstick to manage script complexity), and the block size, which puts an upper limit on how many transactions can be in a block.

Note that transaction validation as an isolated operation is still mostly linear - signature validation has a fixed, but high cost. Validating 2N signatures will take roughly twice as long as validating N signatures. It's just that blockchains naturally tend to grow at an exponential rate, so the number of transactions in block 2N may be exponentially more than those in block N, thus leading to a non-linear validation time.

A blockchain such as Bitcoin that runs at near-full capacity most of the time will still have a linear validation cost for recent blocks.