You are able to upload shapefiles onto CARTO, however they have to be in a zip file with all related extensions.
They provide details in their documentation here
CARTO creates datasets from shapefiles by importing a single zipped file. If your shapefile is not importing, make sure that:
You are uploading a zipped file, and not just one of the files it contains, such as a .shp file.
Your zipped file contains .shp, .dbf, .shx, and .prj files.
Your file names all have the same prefix. For example myshapefile.zip, myshapefile.shp, myshapefile.dbf, myshapefile.shx, and myshapefile.prj.