OK, so here we have local files as origins. In that case, Firefox by default treats each distinct file URL (distinct except for anchors or parameters) as a new origin, thus they cannot share local storage. This is governed by the security.fileuri.strict_origin_policy preference, which is true by default for a very long time now (at least since 3.6, it seems, see bug 539050). Did this ever work for you?
Bug 1730419 Comment 4 Edit History
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OK, so here we have local files as origins. In that case, Firefox by default treats each distinct file URL (distinct except for anchors or parameters) as a new origin, thus they cannot share local storage. This is governed by the security.fileuri.strict_origin_policy preference, which is true by default for a very long time now (at least since 3.6, it seems, see bug 539050).