Is there a maximum number of features that QGIS can save into a shapefile?

Officially, .shp and .dbf files are limited to 2GB. Using GDAL Shapefile driver this can be exceeded, but as stated on this site, this is not recommended due to compatibility issues (See point #Size Issues).

As @Gabriel C. already mentioned, geopackage is a good alternative. You could also use a SpatialLite or SQLite database.

You may have a look at this from ESRI : geoprocessing-considerations-for-shapefile-output.htm

specificaly the part on Geometry limitations:

There is a 2 GB size limit for any shapefile component file, which translates to a maximum of roughly 70 million point features. The actual number of line or polygon features you can store in a shapefile depends on the number of vertices in each line or polygon (a vertex is equivalent to a point).

Shapefile have many other limitations (short field name, no datetime format, 255 character field length limit, problem with null value....) that should motivate you to use other format. You could for exemple use geopackage or spatialite database (both are single file format that are comparable to ESRI file geodatabse, geopackage being a simplified spatialite base)