In terms of artwork and material culture, perhaps. Modern stereotypical "Celtic" art motifs are ultimately descended from La Tene style material culture which featured familiar motifs as concentric circle and dots, repuce in metal objects and more generally a tendency for abstract and ambiguous animal, human and vegetal forms which are often combined. However La Tene style material culture was not confined to speakers of Celtic languages; La Tene style objects have been found throughout temperate Europe in regions that Celtic speakers obviously did not inhabit (La Tene style shields etc. & the famous Gundestrup Cauldron have been excavated in what's now Denmark, which would have been home to Common Germanic speakers).

Why? Simply because La Tene stuff was cool. The majority of La Tene objects discovered aren't your everyday gruel bowls or tableware; they're much more often objects of high status such as swords, shields, cauldrons, jewelry, flesh hooks (associated with feasting) or mirrors. La Tene style objects evidently had some "brand appeal" that led socio-political elites of other linguistic communities to obtain them in order to demonstrate their higher status, just as Gaulish speaking aristocrats would have done. Nor was La Tene art an exclusively "Celtic" art style; it wasn't even purely "Celtic" in its genealogy. Based on an earlier and more conservative Halstatt style (probably innovated by earlier Celtic speakers), La Tene was significantly by Scythian, Etruscan and Greek art in regards to its incorporation of lifelike portrayals of animals & exaggerated lines, detailed figural forms & floral motifs and floral & vegetal motifs respectively. So that's all to say that yes, Celtic speaking people would have had recognizably "Celtic" artwork and objects, but so would speakers of other language groups, and that what we think of as "Celtic" artwork is essentially a synthesis of indigenous designs, Mediterranean and Eastern European motifs.

Linguistically? A Gaulish and Old Irish speaker MIGHT recognize a couple cognate words here and there (curmi- & cuirm 'beer', matu & maith 'good', toncmi & tongmai 'we swear' etc.), but Continental Celtic languages were MUCH more linguistically archaic than Insular ones. That means that they preserved nearly all of the Indo-European system of case inflection that was extremely simplified in Old Irish and completely lost in British Celtic. This means that ironically, Continental languages like Gaulish were grammatically more similar to Latin than Celtic languages like Old Irish, and even shared other linguistic features with the former that are absent in the latter (such as a conjugated form of the verb 'to be' - Gaul. esti vs. Lat. est 'you are'; very similar case endings; similar sentence structure; even loads of incredibly similar cognate vocabulary). By contrast, Insular Celtic languages developed a system of initial mutation that is completely absent in Continental languages, a strict verb-subject-object word order (in contrast to Gaulish subject-verb-object - tho this is less important in Gaulish as it relies more on case inflection) and a loss or simplification of case inflection. What this means is that if you plopped speakers of Gaulish, Classical Latin and Old Irish in the same room without knowing that Celtic languages existed as a genetic group, you'd probably think that Gaulish and Latin were more closely related than Gaulish and Old Irish!

In terms of society, we really can't say much except for what can be inferred about the social dynamics of Continental Celtic speakers from archaeology and what scraps of information we can trust from Classical authors. Both Insular and Continental speakers lived in societies dominated by warrior-aristocrats who drew their power from dependent clients, but that really says nothing about common Celtic cultural features because so did every other early Indo-European society, including Rome. The "clan" system of Ireland seems to have been an indigenous development of a surprisingly late date, intended to secure aristocrats a dependable source of labour and sustenance in a society where wealth was measured primarily by cattle rather than land or money. In Gaul, we see the same pattern of underlying client-patron relations, though in Caesar's De Bello Gallico we hear of Gaulish republics with senates in addition to monarchies, implying that the political dynamics of Continental Celtic speakers were quite varied with no single "Celtic" socio-political structure.

This is all to say that if you went back in time with no prior knowledge of European history, you'd be more likely to draw correspondences between speakers of Continental Celtic languages like Gaulish peoples of the Mediterranean than between Continental Celtic speakers and speakers of Insular Celtic languages.