I think you need to go back a step before investigating Eckington. In spite of it being an often repeated fact, there is no actual evidence that George Soule of the Mayflower was born in Eckington. We have records of a number of George Soules baptised in various parts of England in the right period, but nothing to indicate which if any of these is the right George! He could have come from anywhere – I have even seen it suggested by a member of The Soule Kindred that he was Dutch, not English, though I don't think there's any evidence for this either.

With regard to the two couples you mentioned:
Thomas and Mary (Iddenden) Soole lived in Kent and there is no evidence that they had a son George or that they were connected to the Eckington family.

The Robert and Elizabeth Soule that you mention couldn't have been the parents of George (Elizabeth died in 1566), but this Robert was a member of the Salter's Company. There was another Robert and Elizabeth in Eckington who did indeed have a son George in approximately the right time frame but that George was still in Eckington in 1636 (when he married Susan Nash) so can't be the one you're looking for. (And neither of these Elizabeths was Elizabeth Tylson – the Robert Soule who married Elizabeth Tylson lived in Pinchbeck, Lincolnshire and was part of a completely different Soule family.

On a more general note in medieval London tradesmen could not practice their trade unless they were members of a guilds. These guilds were and are still known as the livery companies and the Salter's Company is such a livery company. They were almost invariably set up by royal charter; they are run by the members and are not owned by anybody. There are two ways to become members of one of these livery companies: either by serving an apprenticeship with another member or by patrimony (i.e. the son of a member of the guild could be admitted to the guild without being apprenticed).