SHA-256 private/public coordinates to the PEM format
What you are sharing is indeed (lapo.it ASN.1 decoder) a 1023 bit public RSA key. RSA keys can be used for multiple things such as signature generation. In that case the signature algorithm may use SHA-256 to hash the message, but the key itself doesn't care about the hash function used (you won't see it in the structure in the link).

&quot;private/public keys SHA-256 keys.&quot;

SHA-256 is a hash algorithm; it doesn't use any keys and certainly not a key pair.
OpenSSL command line only indicates the hash algorithm, but in that case it just assumes PKCS#1 v1.5 compatible RSA signature generation / verification if you use an RSA private or public key respectively; possibly other API's or command lines do the same.

However it returns me not a public key in a .pem format but, according to the datasheet, it returns me the coordinates X and Y of the public key, which are 64 bytes.

X and Y coordinates are not components of an RSA public key. Together they are usually indicating a point on an elliptic curve though. What you are likely getting is a flat representation of EC public key (usually indicated as variable name $W$). In that case they public key may consist of 2 times 32 bytes (i.e. two statically sized unsigned big integer values).

My first wrong assumption can be the fact that I am assuming that a SHA256 pair is the same thing as RSA-SHA256. I know that there are many variants of SHA256 algorithm, but from what I've read the RSA is the standard variant. So I am assuming that RSA-SHA256 is the same thing as SHA256. Am I right?

So no, there is no such thing as a SHA-256 pair. More likely it is a P-256 (or, alternatively named secp256r1) public key. Here P-256 and secp256r1 indicate the curve's domain parameters by name, i.e. a named curve.

Assuming I have the RSA-SHA-256, how can I convert these coordinates to .pem? On the server side, I was using the .pem public key of the device to verify the signature. Now I am only able to share some coordinates of the public key? It is possible to convert from coordinates to .pem? Or am I just misunderstanding a bunch of different concepts?

That's better asked at StackOverflow indicating the exact runtime &amp; libraries that you plan to use.
The steps are:

put the numbers into the X9.62 structure;
put those into a SubjectPublicKeyInfo ASN.1 structure while indicating elliptic curve cryptography as algorithm and the domain parameters by OID;
DER encode above structure;
perform the PEM encoding to apply the &quot;ASCII armor&quot;, i.e. make it text.

Yeah, I know, outside of your experience field, sorry.