Close friends with; sharing confidences.
We might expect ‘as thick as thieves’ to be a variant of the other commonly used ‘thick’ simile ‘as thick as two short planks’. The fact that the former expression originated as ‘as thick as two thieves’ gives more weight to that expectation. As you may have guessed from that lead in, the two phrases are entirely unconnected. The short planks are thick in the ‘stupid’ sense of the word, whereas thieves aren’t especially stupid but are conspiratorial and that’s the meaning of ‘ thick’ in ‘as thick as thieves’.

‘Thick’ was first used to mean ‘closely allied with’ in the 18th century, as in this example from Richard Twining’s memoir Selected Papers of the Twining Family, 1781:

Mr. Pacchicrotti was at Spa. He and I were quite ‘thick.’ We rode together frequently. He drank tea with me.