Questions & Answers about Я слышу красивую птичью песню.
Why is the word for "bird's" птичью instead of having a standard adjective ending like -ую?
Птичью is the feminine accusative form of the possessive adjective птичий. Unlike standard adjectives (such as красивую), these animal possessive adjectives keep a soft sign (ь) in the stem for almost all forms except the masculine nominative, and they take shorter soft endings. Here, it takes -ю, resulting in -ью.
Could I say песню птицы instead?
Yes, using the genitive noun птицы (of a bird) is grammatically correct. However, Russian uses possessive adjectives like птичий heavily for things naturally produced by or belonging to animals. Птичья песня functions almost like a single concept (birdsong), whereas песня птицы sounds slightly more literal (a song belonging to a specific bird).
Why don't красивую and птичью match endings if they both describe песню?
They belong to two different declension groups. Красивый is a standard hard-stem adjective, taking the regular feminine accusative ending -ую. Птичий belongs to the special possessive adjective group, which uses the soft sign (ь) plus the ending -ю in this case. Even though they look and sound different at the end, they are both perfectly matching the feminine accusative noun песню.