Learn German: Dieses Lied ist mein Liebling.

It is perfectly understood, but not the most typical choice. As a standalone noun, der Liebling usually means "darling" or "sweetheart" and is used for people or pets. For objects or concepts, German speakers overwhelmingly prefer using it as a prefix, so it would be much more natural to say Dieses Lied ist mein Lieblingslied.

Why do we use the masculine mein for Liebling when it refers to das Lied, which is neuter?

When two nouns are equated using the verb sein (to be), they both take the nominative case, but they keep their own distinct grammatical genders. Because Lied is neuter, it takes dieses. Because Liebling is always a masculine noun (der Liebling), it takes the masculine possessive mein, regardless of what it is being equated to.

Could I just say Dies Lied instead of Dieses Lied?

Generally, no. When the demonstrative pronoun ("this") sits right next to the noun it modifies, it needs the full ending to match the noun's gender and case (neuter nominative, hence dieses). You can only use the shortened form dies when it acts as a standalone pointing word, such as in the sentence Dies ist mein Lied (This is my song).

How else could I express this idea without using a noun for "favourite"?