Are the names of tapes also saved on the tapes themselves or only in the Veeam database?
Yes. Each tape keeps the meta. For example, if restore should be performed from two tapes you will get a prompt for required media during recovery (either barcode or tape sequence number).
Is there a Veeam tool to read this data without Veeam console?
We dont have such tool, however, Veeam B&R uses the default (unmodified) Microsoft Tape Format. Speaking frank, it should be possible to restore a file from tape by any third party tools that refers to the same tape format (although we cannot guaranty third party tools support).