Is that a bug, or are there hidden songs?
Nope, it's not a bug, and yes, there are hidden songs since v1.1 (before the introduction of chapters system). As of current version (v10), there are 30 hidden songs (including 11 songs from "Cytus Alive", a hidden chapter) + 20 secret charts (same song, different charts, exclusive to Chapter L).
Do I need to pay another IAP?
No, you only need to pay for the chapter itself. After unlocking the chapter, you get access to all songs on it, including the hidden songs.
How can I access them?
As mentioned before, there are 30 hidden songs + 11 "Alive" songs.
For the first 10 chapters (Chapter I - Chapter X) plus Chapter Million, each chapter has a corresponding "Alive" song, which can be accessed by pressing the Easy (green)/Hard (red) circle on the "Chapter Select" screen.
For the rest of hidden songs, you have to do specific thing on the song selection screen. To confirm the hidden song, the ring around "START" text will be colored red:
Ververg (Chapter I)
Tap the torch when the light is at the dimmest point: Ververg ver.B
Precipitation (Chapter II)
Drag two fingers from top to bottom, following the line: Precipitation ver.B
Entrance (Chapter II)
Drag the left wing's shadow to the center: Precipitation at the Entrance
Drag the right wing's shadow to the center: Precipitation at the Entrance II
Saika (Chapter III)
Tap both title cards (彩 & 華) to change the color from red to greenish-blue, then tap to start the song: Saika ver.B
L (Chapter VII)
Hold the title "L" on the center until the color changes to red: L2 - Ascension: Act 1 (Loneliness)
Hold the title "L" on the center until the color changes to blue: L2 - Ascension: Act 2 (Liberation)
∅ (Slit) (Chapter VIII)
Drag from upper-left to lower-right of the shield, mirroring the slash: ∅ (Slit) I
Drag from upper-right to lower-left of the shield, following the slash: ∅ (Slit) O
First Gate (Chapter IX)
Tap the "FIRST GATE" title to display "OVERDRIVE" stamp, then start the song: First Gate OVERDRIVE
FREEDOM DiVE (Chapter X)
Swipe down the screen. The song title will rotate and change to: FREEDOM D↓VE.
Where You Are Not (Chapter K)
Clear the first 8 songs of chapter K (The Way We Were until The Fallen BLOOM) on any difficulty.
Music. The eternity of us (Chapter K)
Clear Where You Are Not on any difficulty.
Hay Fields (Chapter R)
Tap the Libera me's "cross" object: ∞ Fields.
conflict (YM2151 Edit) (Chapter R)
Tap the title "conflict" quickly 13 times until the background becomes not pixelated: conflict
Magnolia (Chapter D)
Tap the red flame above the girl's hand: Myosotis.
L9 : Meteo - Mourn (Chapter L)
Clear the first 8 songs of chapter L (L1: The Devastated Lower until L8 : Meteor - Farewell) on any difficulty.
L10: In Memory of Maneo (Chapter L)
Clear L9 : Meteo - Mourn on any difficulty.
The Ricochet (Chapter M)
Tap the blue spinning disc on the android's head, the background will turn slightly red: The Long Years
There are 20 (10 per difficulty) secret charts which can only be found on Chapter L. On the music selection screen, a letter "L" will appear briefly after changing the song. Tap the location of the letter to access the secret chart1.
L1: the hand (palm) of the boy
L2: the boy's hand
L3: between the wings
L4: the boy on the background
L5: the hand (palm) of the guy wearing red shirt
L6: the hand with the red cross
L7: the shoulder of the girl
L8: the white shirt of the boy
L9: the end of the sword
L10: the center of the sword
Reference: Cytus Wikia
1 original charts on v9.0