One of the things claimed by several adults when I was little is that coffee would stop me from growing so I wasn't allowed to drink it, is there any truth to that claim?
This seems to be more about caffeine than coffee.
In 2008, due to this study that suggested that caffeine intake during pregnancy could increase the risk of fetal growth restriction, the "UK Government's Food Standards Agency" issued new advice to pregnant women on daily caffeine consumption:
Pregnant women are being advised to limit their daily caffeine intake, ideally keeping this below 200mg a day (previously it was 300mg). This is roughly two mugs of coffee a day, although caffeine is also present in tea, chocolate, some soft drinks, and certain medicines.
Too much caffeine might result in a baby having a lower birth weight than it should, which can increase the risk of some health conditions for the baby in later life, or could possibly result in spontaneous miscarriage.
We would emphasise that the risks are likely to be very small and believe our new advice, which is based on new research and has been considered by leading independent scientists, is sensible and proportionate.
As stated above, caffeine isn't just in coffee, it can also be found in (ice-)tea, soft and energy drinks, cocoa and chocolate. I'd guess these sources for caffeine are more common for children than coffee.
Here is a list of caffeinated beverages (including the amount of caffeine in it).
BabyCenter has this to say:
No. Caffeine has plenty of undesirable effects, but interfering with a child's growth isn't one of them.
The United States hasn't established recommended limits for kids' caffeine consumption. In Canada, it's recommended that kids ages 4 to 6 get no more than 45 mg a day, kids ages 7 to 9 get no more than 62.5 mg, and kids ages 10 to 12 get no more than 85 mg.