Portal for ArcGIS has the same limitation when it comes to adding features directly to a webmap, and this isn't necessarily a software related limitation, but a browser limitation. The more individual features a browser needs to render, the slower it will perform. What you'll need to consider doing is creating services to serve your data, rather than render it directly within a webmap. There are a few avenues you can go through:
Uploading content to ArcGIS Online through My Content
You can publish a hosted feature service to ArcGIS Online, which will contain all of your points. If you expect to display more than 1000 features at a time, you'll need to increase the max record count returned by the server.
Publishing content to ArcGIS Online through ArcMap
You can author a map in ArcMap and publish directly to ArcGIS Online as a hosted feature or tile layer. The same concepts above apply for a hosted feature layer.
Publishing content to ArcGIS Online through an app
You can use Esri Maps for Office to publish directly to ArcGIS Online as a feature layer.
You can also publish directly to ArcGIS Server, if you have it installed, so that you can server dynamic services, feature services, or cached services, which can be used in ArcGIS Online as a "hybrid" approach, or Portal for ArcGIS in a full "on premises" type deployment.
Ultimately, it sounds like you need to look into creating a service for your data, whether you host it through ArcGIS Online or Server, and how you actually get it there, is up to you.