This can apply to a couple of different areas.
Face value as it pertains to the financial world is the nominal, or dollar, value of a security. With a stock, it is the original price it was sold for as printed on the certificate. For bonds, it is the amount paid to the holder at maturity. This is, in the bond market, called par (or par value).
When we start getting into currency (along with stamps) is the amount printed on the individual coin or banknote. This is set by the issuing authority. For this market part, this is considered the legal value. Market price can be different as in the case with coins and collectors.
We have to remember that most asset in the financial world were created when we dealt with things physically. The move away from paper to digital is now about 30 years old. This will continue to expand as payment applications become more popular, digital wallets are embraced, and financial products are brought out exclusively in the digital realm. The latter is going to be greatly influenced by decentralized finance (DeFi).
One reason why face value is important is to determine if a security is being offered at a premium or not. This is an important part of the bond market. When interest rates are bid up as compared to the coupon rate, the bond is at a discount. The reverse is true where a bond is at a premium if the interest rates is below the coupon.