That is one important thing to consider. I would recommend people naturalizing to just keep old mobile number and contract for a while. Then make a new contract with a new mobile phone number with the new Taiwan ID number.
Then switch over accounts or even register completely new accounts, closing the old ones.
Quite a few banks, businesses use different systems that connect via ID number and those are not designed to be changeable. This often breaks features and connections to other things as the ID on the account does not match. e.g. bank to Line account connection
I experienced this with Easy Wallet that connects to my bank accounts. Once I got the new ID number, the connection between easy wallet and bank account error out. I had to contact them both to update the number, and easy wallet had to manually update the connections as I was not able to remove/recreate them. This took them many moths to fix.