In addition to the legacy already mentioned, it's intuitive for the spacebar to not have any label because it produces an empty space. In other words, an 'invisible' label is more representative of an 'invisible' character.
Why there is no symbol on the space bar?
A space is just… space, with no visible character. The spacebar shows no visible character because that's exactly what it produces when you press it — a space, with no visible character.
How do you write a space bar?
Should I use "space bar" or "spacebar" in my writing? When referring to a key on a keyboard, both "space bar" and "spacebar" are correct. At Computer Hope, we use the single word version "spacebar" or "spacebar key" in all of our writing.