Two versions

Info, and a correction.

When you pay for an Adobe Plan, when you pay for an subscription. What you get in terms of applications depends on the Plan you purchased.

If you went for the Lightroom Plan, all you would get is Lightroom (LR) And that is the flavor of lightroom that runs on your computer, but saves a Album in the cloud, and stores photos on the cloud. It does not use a catalog (it accesses an Album, on the Cloud). Adobe is starting in support documents to be calling this Lightroom Desktop. Some members refer to it as LR Cloudy, IMO should be called LR POS. The Lightroom Plan comes with 1TB of cloud storage.

If you went for the Photography Plan, then you get rights to use Photoshop (PS), Lightroom (LR), Lightroom Classic (LrC), Bridge, and some others. The Photography Plan comes in two options. the 20Gb, and the 1TB cloud storage.

Additional plans exist, wont get into them.

Nothing wrong with having both LR and LrC installed,

The Photography Plan contains both Lightroom Classic and Lightroom (cloud). The latter as a kind of trial option, to get an idea what Lightroom (cloud) is all about. There is also another reason: if you sync images from Lightroom Classic to the cloud, they are synced as smart previews and do not count against the 20 GB cloud space. That means that you still have the option to sync, for example to showcase your images on an iPad or iPhone. Many Lightroom Classic users do just that. You can even edit images on your iPad,  but there is a caveat. Any custom profile you use in Lightroom Classic will show as a missing profile on your iPad, because profiles don't sync. Here Lightroom (cloud) comes into play. You can consider this as "Lightroom Mobile on your Mac or PC", with one extra option: Lightroom (cloud) can import these custom profiles and then they will be synced. Same for presets you create in Lightroom Classic and would like to use on your iPad too. You can import them and sync them via Lightroom (cloud).

If you first downloaded and installed the Cloud Centric version use the CC Desktop App to Uninstall it.

Please Open Both programs, one at a time, and then go to Help System Info and post the first 5 lines from each of the dialog boxes that come up.

With the Photography subscription plan you are allowed to install both LrC and Lr (LrC is the Classic, Traditional version and Lr is the newer Cloud Centric version that uploads all your images to the Adobe cloud storage system) installed on your computer but you still only get 20Gbs of cloud storage space.

We need to know which exact versions you have installed.