AFAIK, "E.I.G." was a pre-1968 US importer of mostly utility grade firearms from various sources, many in Italy, like your shotgun.
While some shotguns were made by a single entity, and usually also bear a maker's mark (Your pics don't show enough detail in all the correct areas, where the stamps were usually applied) - many weren't marked because they weren't made by a single entity.
Gun, mostly shotguns (and especially Italian shotguns & some Spanish), that weren't stamped with a maker's mark usually were made through a consortium of gunmakers in the Brescia region of Northern Italy, in the Gardone Valley.
Since they were all members of the Italian Gunmaker's Guild (a little like a US labor union) - and most of them made what were basically clones of a single O/U design that's come to be referred to as a generic inexpensive Italian O/U.
Guns made by them are simply referred to as "Guild Guns".
The separate entities/companies may or may not have been family-related, but typically one entity made the barrels, another the action, another the stocks, and yet others who fitted & assembled everything & forwarded the completed gun to some wholesaler for export.
Soooooo, your gun may have a minute maker's mark somewhere, that you're not seeing, or don't recognize for what it is - or it's a "Guild Gun".
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