How to know if a public key has been created based on the ECDSA algorithm?

to create a valid identity, a user must generate an ECDSA public key and then send it to the network administrator for validation

That's under-specified, lacking:

• The Elliptic Curve used. Common ones are in sec2v2 section 2, including secp256r1 (used in many commercial and e-government applications) and secp256k1 (used in bitcoin and several other cryptocurrencies).
• The format for the public key. Common variants include with or without point compression, optional ASN.1 (typically DER) decoration/encapsulation, and then optional binary-to-text translation.

With this, there is a well-defined process to verify that the public key is valid. First check that it is per the specified format, and peel that as necessary to perform Validation of Elliptic Curve Public Key, and within this the Elliptic Curve Public Key Validation Primitive (notice that depending on the public key format, it might be necessary to first perform point decompression as in Octet-String-to-Elliptic-Curve-Point Conversion).

Update: A common public key format in bitcoin (after 0.6) is the raw compressed public-key format for the implicitly specified curve secp256k1. A validity check of that boils down to:

• Check that the public key is exactly 33 bytes.

• Check that its first byte is 02_h or 03_h (this byte codes the parity of the $y$ coordinate, and needs no further check).

• Convert the remaining 32 bytes to integer $x$ per big-endian binary convention, which implies $0\le x<2^{256}$.

• Check that $x<p$, where $p$ is the prime $2^{256}-2^{32}-977$.

• Compute $s\gets(x^3+7)\bmod p\,$

• Check that $s^{(p-1)/2}\bmod p\,=\,1$. Per Euler's criterion, this verifies that there exists integer solutions $y$ to the curve's equation $y^2\equiv x^3+7\pmod p\,$. On curves with cofactor $h=1$, including secp256k1, this proves there exists a matching private key.

  Note: Sometime we need the Cartesian coordinates of the curve point defined by the public key. Since $p\equiv 3\pmod 4$ for secp256k1, that can be done efficiently together with a slightly modified version of the above last step:

  • Compute $y\gets s^{(p+1)/4}\bmod p\,$.
  • Check that $y^2\bmod p\,=\,s$, which completes the check.
  • If the low order bit of $y$ does not match the low order bit of the first byte of the public key, then change $y$ to $p-y\,$. Now $(x,y)$ are the desired Cartesian coordinates, with both $x$ and $y$ in $[1,p)$.

The above tells how to check that the public key is valid, but not that the user (or anyone) knows the corresponding private key. This is best done after validation of the public key (in the above sense), then by the challenge/response method in the question.

Checking that the user knows the corresponding private key has one advantage: if the "text to the user" is unique to each key validation, and can't be confused with other messages that the key will sign, then this prevents a user from registering the preexisting public key of another person (which private key is secret).