It is Latin. Caligula is a diminutive of 'caliga', the standard Latin word for a soldier's boot.

The wider implications of your question are really interesting though: what other languages were spoken in Rome in the 1st century? I imagine that Greek would have been the next most common after Latin, especially among the slave population who were often imported from places further east. Perhaps they also brought some other eastern languages too - there seems to have been a Syrian enclave in Trastevere for example, and there may have been some Demotic speakers involved in the grain trade from Egypt. There were certainly Jews in 1st-century Rome, so you might've caught the odd snatch of Hebrew too. Gauls and Germans would've been present in the city too, so their languages might've been heard as well.