Yes, this ~とかけて~ととく is a traditional joke pattern known as 謎かけ. See: Meaning of sentence
This unusual と is a variant of a quotative-と. See: Function of と when used with 続く
In these lyrics, two phrases (ひとり手酌のお酒 = a bottle of sake in my hand; 橋の通わぬ川 = a river that no bridge crosses) are presented in a riddle-like fashion, followed immediately by its "answer". However, the standard translation of "What do A and B have in common?" doesn't seem to be very useful in interpreting this song. Instead, in this song, let's consider it as a clichéd expression that vaguely hints at the relationship between the two phrases.
The "answer" part is:
渡りきれない あなたの胸に 酔えば切なくなるばかり
Here, 渡りきれない ("uncrossable") is clearly related to the river, and also can be understood as a metaphor for "your heart being unreachable".
All in all, the song is basically saying "While drinking sake alone, I realize how unreachable your heart is, like a river with no bridge, and this only deepens my sorrow", but in a tricky nazokake-ish fashion.