Why do these “inverted indexes” just look like indexes to me? Too much time with databases perhaps?

The distinction is more clear when indexing actual text and applying tokenization. A "typical" index on a database column goes like "column(value => rows)". When people mention inverted indexes its usually in the context of full text search, where "column value" usually goes through tokenization and you build an index for all N tokens of a column "column:(token 1 => rows)", "column:(token 2 => rows)",... "column:(token N => rows)".

A non-unique index, yes.