How are bip44 coin identifiers assigned? There is a list of bip44 identifiers for altcoins here.

It uses the term registered coin types.  My question is, is there anything in code for each coin that the bip44 number is derived from, or this is purely an arbitrary assignment from a 3rd party?

From reviewing code, bip44, and slip44, my conclusion is the latter: namely that one "registers" simply by sending an email to the author(s) of slip44, and they hopefully assign a number.

If I am incorrect and there is code in bitcoin-core or an altcoin derived from it that defines or uses the bip44 id, please point me towards that code.

thanks!
BIP44 is more like a standard which you are supposed to follow. The coin id's assigned are way to standardise the approach in case you are making a wallet and want better segregation of your coins.

For example, with the same seed, you can derive keys for different cryptocurrencies, just by changing the coin  id. Do note all those ids can be used to generate addresses for only one crypto as well.

Basically it is trying to maintain some nomenclature in this open source community. There is no code in the source code specifying this.
As a matter of fact , the address and private keys generated are backwards compatible to a node version not having the HD wallet implementation.

Address generation is something which the bitcoin core software provides, however anyone can write their own code for address generation as long as it follows the guidelines for address generation.