Cualquiera literally means "anyone" or "whichever". However, in this song, it's used in a derogatory way to mean "a nobody" or "a tramp".
In the lyrics, the singer tells the woman, "solo eres para él una cualquiera" (you are just a nobody to him). This powerful and somewhat harsh usage highlights the painful reality the woman faces, making it a very impactful and memorable word in the context of the song's narrative.
Fidel Rueda's "Aunque Te Duela" is a blunt, no-nonsense song about a man trying to open a woman's eyes to the truth about her lover. He sees her pain, noticing "que lloras tú por él" [that you cry for him] and that "te volvió a fallar no vino otra vez" [he failed you again, he didn't come again]. He's frustrated that she's letting this man "arruine así tu vida" [ruin your life like that].
He doesn't hold back, telling her directly that her lover is just playing with her and only seeks passion, not love. The most striking part is when he reveals the harsh reality: "Y aunque te duela solo eres para él una cualquiera que reemplaza en la cama a la mujer que está en su corazón" [And even if it hurts, you are just anyone to him, who replaces in bed the woman who is in his heart]. He insists she's just "un pasatiempo un juguete en su vida" [a pastime, a toy in his life] and tries to convince her, "tú tienes que entender que soy mejor que él" [you have to understand that I am better than him].