This is an interesting question and one that does not have a uniform answer.
For example, in Colorado, while there is a court proceeding by which you can change your name, there is also case law precedent that says your legal name is what you go by even if it doesn't match your birth certificate or other official records.*
For example, I have a client who has a birth name completely different from the one she uses in day to day life (as different as, for example, "Sophia" and "Betsy" are from each other) that she picked up in college and just started living under for reasons that she could never really fully explain to me in a way that made sense. Under Colorado law, the name she used on a day to day basis is one of her legal names, even though there was never a legal name change proceeding conducted in a court.
I happen to know someone who set up a bank account with a name other than on their official documents and with their own Social Security number at a bank and it caused no problems, because banks don't routinely check Social Security numbers against names (employers, in contrast, routinely do cross-check).
Not every state recognizes "common law" name changes, or even every bureaucratic entity within a state that expressly does have that as their law. Most bureaucrats in both the public and private sector don't understand the concept of a common law name change even in places where that is the law, because it doesn't fit with bureaucratic logic.
It is not forgery, which would involve signing with an intent to appear to be some other actual person who you are not.
Whether or not it is fraud turns on intent. If you use a different name to evade a creditor or to cash a check that wasn't intended to be payable to you, that is fraud. But, simply changing a new to avoid family members who mistreat you is not fraud.
In terms of getting the bank records, the best thing to do is to try to explain the situation to the bank official (ideally with, e.g., a Social Security card and other evidence such as the name of a former employer or a person to whom checks were regularly written to corroborate your story).
Name disconnects aren't that unusual in banking. For example, I was beneficiary of an account established by my late father that identified me by my pre-marital name of 23 years ago. But, the rule of law is that a beneficiary designation is valid if the intended person is clear, even if the names don't exactly match, for example, because your name changed in the meantime. As a result, I am able to deposit checks payable to my pre-marital name. So, a lack of a name match shouldn't be a strict bar to getting old banking records of your own account.
For another purpose, it is also permissible to have checks whose name don't match the name on the account, and to have "trade name" accounts of an unincorporated individual supported only by an obscure trade name filing for businesses.
Post-Script: Some Similar Concepts
- In the same vein, Colorado recognizes common law marriage that arises when two people agree that they are married and hold themselves out to the public as married, without any formal paperwork. If you just start calling someone "my husband" and you both mean it, it is true.
Emancipation of a minor in Colorado is a similar concept. An emancipated minor in Colorado is a minor who provides most of his or her own support and lives independently. Courts in Colorado don't declare that a minor is emancipated causing the minor to become emancipated following a petition requesting emancipation. Instead, courts in Colorado determine as a factual matter if given the way that the minor lives his or her life, he is or is not emancipated. The emancipation determination of a court flows from the reality on the ground, rather than the other way around.
The race-notice recording system for real estate also works this way. Ownership of real estate flows from transactions that happen outside the recording system and can be private. Recording real estate documents provides evidence of an underlying reality and provides a means of weighing evidence when there is a dispute over who owns real property, but recording a document with the clerk and recorder of a county in not what actually causes title to that real estate to change, unlike the situation in title registration jurisdictions like the UK and Germany and South Africa where the filling with the government office is the moment at which title is changed.