If you look at the list of Standard Languages, you'll see that most of them use Dwarvish as their script:
- Common: Common
- Dwarvish: Dwarvish
- Elvish: Elvish
- Giant: Dwarvish
- Gnomish: Dwarvish
- Goblin: Dwarvish
- Halfling: Common
- Orc: Dwarvish
(While that's 5e, I found a similar chart for 3e.)
The Forgotten Realms Wiki calls this script Dethek and explains some of its history, but I can't see how the explanation there relates to the Standard Languages at all. I also know next to nothing about DnD in-universe history.
Is there an in-universe explanation for as to why these languages all use the same script?
I don't particularly care where the explanation comes from as long as it is from some DnD canon where the premise is true. I don't expect too many settings to have an explanation at all, which should keep this from being too broad.