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Long answer:
The highest a stat can go is 20, unless the racial starting value is higher than 8.
The way that works is, you can invest max 12 character points in a stat, and then if that value is lower than 20 after that, it can still be increased with buffs like spells, items, consumables.
So for example, a human starts with 8 Dexterity, and can go up to 20. An elf starts with 9 Dexterity, so their max is 21. On the other hand, a dwarf starts with 7 dexterity, so they can only get up to 19 with character points, but can still reach 20 if they use a dexterity buff like the agility of fire spell or wearing a charged ring.
Dog's starting Dexterity is 8, so 20 is the highest it can get, with or without buffs.
You might be thinking of Strength there, since the damage bonus indeed doubles at 20 and above.
For Dexterity, each point is a point of speed, with the exception of Dex 20 at which you get 5 speed extra. It goes something like:
8 Dexterity -> 8 Speed
9 Dexterity -> 9 Speed
10 Dexterity -> 10 Speed
(...)
18 Dexterity -> 18 Speed
19 Dexterity -> 19 Speed
20 Dexterity -> 25 Speed
21 Dexterity -> 26 Speed
A bonus from background can let you get to 20 (or closer to it if your race starts with a penalty), but doesn't count as a racial bonus.
For example half-ogres start with 4 intelligence, and their normal maximum is 16 (4+12).
If the half-ogre has a miracle operation background, he starts with 7 intelligence and can go as far as 19 (7+12).
But if a human has a miracle operation background, they start with 11 intelligence, but their maximum is still 20.