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 - English Children's Songs - England - Mama Lisa's World: Children's Songs and Rhymes from Around the World - Intro Image

Notes

*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.

Comments

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.