No, exporting to shapefiles leaves the attachments behind. Shapefile is an old file format that dates back to the 90s, attachments weren't part of ArcGIS until the late-00s, so shapefiles have no concept of attachments. You need to export to a GDB format to maintain your attachments.
One alternative to this would be to export to a shapefile then write a script (using Python for example) to extract attachments using the REST API, however this is obviously much more complicated than just exporting to GDB.
It's probably worth mentioning that shapefiles are "lossy" in other ways as well, for example date/time fields in shapefiles don't support time components, so you'll lose that information. They also don't handle null values well, Global IDs, etc...
