I suspect it's actually Microsoft Office with Access and a mature Excel that did it. Everyone was using office at the time (for Word, Excel, Powerpoint and the emerging Outlook), so access was "free"; and many things that dBase&co were used for, were actually becoming easier to do in Excel.
Oh man, you just wet my eyes! What a trip into the past!
I wrote my first paid project in dBase in 1982 when noone knew what "computer mouse" was. I answered the ad in newspaper and initially they wanted me to help them organize text files. I told them about dBase and that was my first project - a simple database to manage their parking lots. It was such early days of programming, we had basically zero agreement! Just half piece of paper that they pay me 10% in advance and short list of features to get remaining 90%. And funny thing is - no copyright agreement! So while I made few months of rent for their project, I ended up paying off my first mortgage by just offering the same piece of software to many other parking corps! It was incredible feeling! Like creating money out of thin air - they sponsor my plane/hotel and I just told them what PC (with DOS) I need to be ready. Then in just few hours of installation and training, I was on my way home with nice check. I frankly don't even remember how much it was back then...
Good times!
I suppose the secret is quickly delivering software that does what someone wants.
I was surprised that databases are contained in *.dbf files, and no real database-service is present to process database-calls.
A shapefile is actually a collection of files, typically using the same filename with different extensions.
There is a .shp file containing the actual geographical "shapes": points, lines, polygons, etc.
Alongside it is a .dbf (dBASE) file with information about each shape. For example, if you have a list of cities, each city may have a boundary polygon (or multipolygon) in the .shp file, and the city name, population, and other demographics in the .dbf file.
The other required file is a .shx containing indices into the .shp file for faster navigation through it.
There may be other files as well; a common one is the .prj file that describes the projection used in this shapefile.
This collection of files, foo.shp, foo.dbf, foo.shx, etc. is referred to together as a "shapefile".
https://en.wikipedia.org/wiki/Shapefile
Bringing us full circle, GIS programmers and users often import a shapefile into PostGIS for processing. And PostGIS is, of course, a GIS extension to PostgreSQL!
Funny enough, two popular contenders are built on SQLite.