Is it true that orange, silver, and month are the only three English-language words without any rhymes?
No. Other common English words without perfect rhymes include "purple," "pint," "wolf," and "dangerous." There are others, especially if you allow very long words like "discombobulate."
Note that this applies only to having other single words that rhyme perfectly, excluding proper nouns. If half-rhymes are allowed, then "lozenge" comes close to "orange." If combinations of words are allowed, then either "door hinge" or "car hinge" can rhyme with "orange," depending upon your dialect. Similar solutions can be found with other such words.
You might find our Glossary of Poetry Terms of interest.