I have figured out the difference between a Tenor and an Alto horn.
"Alto horn" refers to an Alto Eb horn in the saxhorn family, while "Tenor horn" refers to the Bb variant of the Wagner tuba or the Tenor Wagner Tuba.
Tenor Tuben was used during the Late 19th Century and Early 20th Century, while Tchaikovsky wrote this in 1880, around 20 years after the invention of the Tenor Tuben.
It's doubtful Tchaikovsky wrote eight parts for just one instrument, the Alto Horn, and named them differently as Alto Horn 1-4 and Tenor Horn 1-4.
I haven't heard of any instrument other than the Eb Alto Saxhorn that goes by "tenor horn." Additionally, the Tenor Tuben is in Bb, which is consistent with the notation shown in the sheet music.
In conclusion, I think that the Tenor Tuben is the instrument that is called upon in this piece of music.