This rhyme could be referring to drinking. "Old father Grey Beard" referred to a pitcher of spirits. You can measure drinks by using your fingers. The thumb is used to grip a mug.
Old father Grey Beard*,
Without tooth or tongue;
If you'll give me your finger**,
I'll give you my thumb.
*Greybeard can refer to an old man or a pitcher for holding spirits.
**Finger – this could be referring to a way of measuring drinks – when you eyeball it, instead of using exact measurements – you measure on the glass with your two fingers held at the bottom of the glass. A person can ask for two fingers of spirits. Two fingers is roughly equal to a jigger which is 1 ½ ounces.
This rhyme goes back to the 18th century.
Thanks and Acknowledgements
This rhyme is from The Little Mother Goose (1912), illustrated by Jessie Willcox Smith.