No, you are correct. Most Mandalorians do have names...
“The character is very much built on the iconic presence of the Man
With No Name in the Sergio Leone movies, played by Clint Eastwood,
[and] the lone samurai in Akira Kurosawa,” he said. “It’s
aesthetically and very, very much narratively built in that kind of
iconic lone gunslinger/sword-wielder.” -Pedro Pascal
As mentioned in a comment, actor Pedro Pascal had let his character's birth name slip in an early interview, but it had yet to be revealed on screen and/or with a lack of clarity that the character has kept his birth name, but the reveal of it implied, that while the series would intentionally play with 'The Man With No Name' allusions, it wouldn't do so forever...
Episode 1.08 (Redemption), the season one finale, changes that. His name is
Din Djarin. This is the same name Pascal let slip prior to the series start.
In addition, looking at this from an in-universe perspective, one reason Mandalorians might not as freely be giving out their names, is because in this era they are on the verge of extinction and are in hiding.
1.03 The Sin Transcript:
The Armorer: It will draw many eyes. These were cast in an Imperial
smelter. These are the spoils of the Great Purge. The reason that we
live hidden like sand rats. Our secrecy is our survival. Our survival
is our strength. Our strength was once in our numbers. Now we live in
the shadows and only come above ground one at a time. Our world was
shattered by the Empire, with whom this coward shares tables.
The Mandalorian: The Empire is no longer. And the Beskar has returned.
The Armorer: When one chooses to walk the Way of the Mandalore, you
are both hunter and prey. How can one be a coward if one chooses this
way of life? Have you ever removed your helmet?
The Mandalorian: No.
The Armorer: Has it ever been removed by others?
The Mandalorian: Never.
The Armorer: This is the Way.
Episode 1.08 reconfirms these ideas that were laid out in 1.03 when the characters come across
a pile of armor and Djarin talks once more with The Armorer.