Anonymity means keeping your name off the Internet. That doesn’t mean you can keep your name from vendors, utilities, banks, etc. Those folks are supposed to keep your information confidential, so you really don’t have a worry there. Your worry is the Secretary of State that may publish ownership information of your LLC, as well as the county records that publishes ownership information of a piece of property.
The same thing goes for the IRS relating to taxes. In order to obtain a bank account, the bank is under federal law to “know who their customer is”. When opening a bank account, the LLC will own the bank account, but if you are going to be a signer, then the bank will need to know about you (and will ask for your personal identification). Most banks require you to actually show up in person at a branch, and I’m not aware of any banks that will do this remotely, over email or fax. Fortunately, the banks are required to keep your information confidential.
As it relates to the IRS specifically, in order to get a bank account in the name of the LLC, your LLC will need a FEIN number. In order to get a FEIN number for your LLC, you will need to specify who the “responsible party” is. Read more from the IRS. The quick summary is that a “responsible party” must either be an owner or someone in control of the company, and must already have a SSN (US Citizen), FEIN (Another company), or a ITIN (non-US citizen).