What is the relationship between the ether value and tokens on the Ethereum blockchain? Are the two intermingled?

ERC20 tokens run on the Ethereum platform. So, any transaction of ERC20 token consumes processing power on the Ethereum platform (or miners mining ETH)

As you must know, processing power costs gas which is paid for, with Ether.

So depending on the number of ERC20 transactions (Not volume), Ether will get consumed accordingly, which could spike its demand and therefore its price.

It’s hard to say whether a success of an ERC20 token will directly impact Ether prices, but the two are definitely correlated and indirectly affect each other.