I have noticed Facebook identifying the Eris differently throughout having it. With 1.5 it was called the
Droid Eris. When i loaded the leaked 2.1 update, and started using ADW it was identified as the
Nexus One (I always thought it was because of ADW), then later on it switched to the current
Desire ADR6200.
ADR6200 is the actual model number of the Eris.
Regardless the facebook app is the same no matter which device Facebook thinks you have as the link will point you directly to Facebook in the marketplace.
The Desire was the original name for the Eris before they changed it (i also think it's funny - deSIRE / ERIS). If you've ever seen the filename for the 2.1 OTA upgrade zip
(OTA_Desire_C_Verizon_WWE_2.36.605.1_1.17.605.1_release_0420.79112f65) and the subsequent July 2010 OTA update zip
(OTA_Desire_C_Verizon_WWE_2.37.605.4_2.36.605.1_release) you'll notice that it says Desire in it.
Why they changed it i don't know, but the Desire became the name for the an HTC Android phone released in the UK.
In a way i guess the Eris IS part of the Desire series of handhelds by HTC.