The amount at stake for identity theft is what determines how hard it is for someone to steal your identity. It does happen quite often, in 2016, 6.15% of adults reported some kind of identity theft. Most of this identity theft was fraudulent credit card purchases, which is one of the reasons credit card chips were introduced.
I'd never had my credit card skimmed until 2008, and since then I've had to get 4 cards reissued. Once someone has your name and number and cvv or zip code, they can easily make gas station and restaurant purchases with fake or reprinted cards. Any store that doesn't use the chip is vulnerable to this.
The next step up in fraud is a little harder, opening a credit account for someone else. You need a proof of age, SSN, address and name. The key protections are your SSN and proof of age. The proof of age either requires a fake ID or a store, who's employee is negligent in application verification. It's good to protect yourself by freezing your credit reports, if you need a credit check, let them go ahead and check against your frozen account, they'll call you, tell you which service they're using and you can unfreeze that service for a window, saving you the hassle of freezing and unfreezing all 3 services for every credit check.
Buying a car as someone else might be a little easier than bank fraud. The due diligence of the financial officer of a car dealership isn't super strict. They'd require the same information as a credit check, but on the bright side, the protection is the same.
Getting into the bank is even a little more difficult, all they do is take money and security seriously. It is difficult to get into the online bank, because they're security locked to your approved cookies, e-mail and phone numbers. Even so, it's possible to socially engineer someone's e-mail address away and reset their password and approve a new cookie if they know what they're doing. Banks are more savvy about checking IDs, and there's a human component to the physical bank, but someone can still get by with a fake ID and a SSN. These are more targeted attacks, and sometimes you need to know more information, like answers to security questions, ability to get a fake ID and knowing a likelyhood that the person isn't living paycheck to paycheck. What's much easier to do is a check fraud scam: to respond to an ad, forge a cashiers check for more than the item is worth, demand cash back (like maybe a payment to a mover for the item).
To protect yourself from bank fraud, make sure your bank and on-file email have multiple check authentication. If you're using your phone for protection, make sure it locks when you're not using it. This will help to prevent phone theft from becoming a gateway to your email and bank accounts. Don't agree to deals that seem too good to be true and never pay out before a check clears.
A fraudulent mortgage would be very difficult to pull off. The bank verifies your information to an extreme measure. The title company verifies the integrity of the property. The thief obviously couldn't keep any large property as a result of the fraud and would have to find some way to take ownership of the money. I've read reports of people stealing identities of a homeowner and go through the process of selling the home, they still pay off the initial mortgage (which I assume is very hard to get around) then pocket the rest.
To protect from mortgage fraud, as the buyer, they won't take your money without a credit check, so freeze your accounts. As the seller, it's not a legal sale if you own the property and didn't sign.