Does the brain treat sign language like speech?

Largely yes: the brain handles sign language much as it handles speech, with a few differences that make it more interesting. The left side of the brain is the dominant side for producing and understanding sign language, just as it is for speech. Signers with damage in Broca's area have problems producing signs, and those with damage in Wernicke's area have problems understanding signed language.

That finding was once a surprise. The left hemisphere was described as the 'verbal' hemisphere and the right as responsible for spatial tasks, and that split was used to denounce signed languages as not equal to spoken ones. Then, in the 1980s, researchers examined deaf patients who had suffered a left-hemisphere stroke. They found that the left perisylvian region was functionally critical for language, spoken or signed. The idea that language needed auditory input, once used to discredit signed languages as 'real languages', fell away.

There is a charming detail. Hearing people have slips of the tongue after left-hemisphere damage; deaf signers have 'slips of the hand', which usually involve an incorrect handshape in the correct location with the correct movement, like a hearing patient saying 'bline' for 'fine'. The mistakes show that the parts of a sign are real building blocks in the brain, too.

The picture is not identical to speech. Processing ASL extensively recruits right-hemisphere structures too, and brain recordings suggest syntactic processing in ASL is not lateralized to the left hemisphere only.