In the first episode of The Mandalorian, the Mythrol appraises the Mandalorian's ship:
Mythrol: I like your ship. She's a classic. // Razor Crest, am I right? Pre-Empire? // I have a lot of credits, by the way. That's why I offered to hail a Cruiser. (transcript source)
There's no indication that he knows who the Mandalorian is or has heard of his ship before, but he identifies it as "Razor Crest, am I right?" implying that it's a type of ship. Similarly, in Chapter 6: The Prisoner, Mayfeld makes another comment about it connecting "Razor Crest" with the ship's age:
Mayfeld: Razor Crest? I can't believe that thing can fly. Looks like a Canto Bight slot machine. (transcript source)
However, the same episode and some official tie-in material seems to use "The Razor Crest" as the ship's name:
Zero: The Razor Crest is scrambling our signature, and I'm inside the prison system. It's impressive that this gunship has survived the Empire without being impounded. (transcript source)
Razor Crest
A former military craft used to patrol local territories in the time before the Empire, the Razor Crest is the Mandalorian's trusty transport and living quarters, as he scours the Outer Rim for fugitives to catch. (starwars.com databank entry)
The Lego model italicizes it on the box as well:
Is there an official source that clarifies this one way or the other? To me it seems most likely that the ship wasn't supposed to have a name and that the characters are referring to it as "The Razor Crest" the way that people in real life might refer to "the Winnebago" or "the 737", but it's also unusual for a Star Wars ship to have no name, especially one that receives so much focus throughout the series.