And if not, how are they classified ethnically and linguistically? It seems that there has been debate in the past, and I am curious what the modern view of the Picts place in history is.
I can't speak to Pictish culture or ethnicity, other than to say there is no meaningful ethnic conception of Celtic, which can be a bit confusing since it is almost always used in popular parlance in a quasi-ethnic context. However Celtic is necessarily a linguistic identity, which is to say it identifies speakers of a Celtic language. The topic of Insular Celtic-speaking peoples and their relationship with the Celts of the Classical world cam up recently, and in particular a response from /u/amusiclistener: "Did the Celtic peoples "arrive" in Britain, or did Celtic culture just spread to the bronze-age inhabitants of the island?"
So. Linguistically — were the Picts Celtic-speaking? With the caveat that I can only summarize the technical opinion of my betters, the decipherable evidence (which is very fragmentary) suggests they were, although there may be some linguistic substratum (perhaps non-Indo-European) involved that confuses the issue. Pictish had previously been regarded as anywhere from non-Indo-European to vaguely Gallo-Brythonic...ish; or in Kenneth Jackson's The Pictish Language, both! However the tide of opinion has turned generally (if not entirely) towards interpreting Pictish as Brythonic, which is to say related to the other Celtic languages spoken on Britain (from whence modern Welsh, Cornish and Breton derive, joining a host of extinct languages such as Cumbric, Invernic and likely plenty more we have no attestation of). While this is hardly a novel concept, being that the Picts co-inhabited an Island with Brythonic speaking neighbours, the actual categorization of Pictish is quite technical to put it mildly†, and involves the appearance of various grammatical and phonological elements between Pictish and Welsh.
Fortunately some of the more overt evidence towards Pictish being a Celtic language is also the easiest to grasp: That being naming similarities between the languages. The majority of attested Pictish forms, or derived forms are toponymic or onomastic, so this is a compelling similarity. Katherine Forsyth offers numerous examples from which I can provide a brief selection: Place-name elements such as lanerc, 'grove' (Welsh llannerch) or pert, 'bush' (Welsh perth), and personal names such as Taran (Welsh taran, 'thunder'), Onuist (Old Welsh Unust), Naiton (Old Welsh Neithon), Drosten (Old Welsh Dristan), and earlier personal naming structures which appear quite Celtic in character - Argentocoxus, 'Silver-Leg' being a favorite of mine. Notably the names for the Picts amongst their neighbors (Prydyn and Cruithin in Welsh and Irish respectively) both derive from terminology for Britons, so they were at least regarded as part of greater Celtic Britain, though geography may have informed that as much as linguistics to be fair. Further a not insubstantial number of Pictish inscriptions are outright undeciphered, and some Pictish forms are demonstrably non-Celtic, so there can still be said to be some uncertainty.
A more thorough treatment of this topic could be found with A Pictish Panorama: The Story of the Picts (1995), Katherine Forsyth's "Literacy in Pictland" in Literacy in Medieval Celtic Societies (1998) and for historiographical context Kenneth Jackson's aforementioned "The Pictish Language" in The Problem of the Picts (1955).
^(† As the usually coherent Katherine Forsyth illustrates in a summary of some arguments made by Celticist John Koch:)
^(There is a dearth of examples showing Pictish turning the voiceless geminate [long or double] stop consonants [ pp , tt , kk ] into simplex spirants [f, q , c ] as in Welsh, Breton, and Cornish, but it seems to have treated them more or less as Irish did; see Jackson, Problem of the Picts 160. Pictish apparently did not lenite [soften] voiced stops preceded by liquids [ r and l ], as in, e.g., the place-name element carden ‘thicket’ [ Problem of the Picts 164]. Several early attestations suggest that Pictish had the o-grade abbor ‘river mouth’ [< Pictish king-list [1444] ad-bhor -], perhaps alongside the e- , whereas British shows only the e - [OW , Bret. aper < ad-bher -: Apur- nethige , Apur-feirc [Pictish king-list , ‘Popple- ton’ or A MS]; stagni Aporici , stagno Aporum [ Vita Columbae 83b, 67b], Aporcrosan [ Annals of Ulster 672], Abbor- doboir [Book of Deer f. 3a].)
^(And on for five more paragraphs. To paraphrase Bill Bryson: No arguing with that! No understanding it either. Also formatting linguistic notation in markdown makes me want to set myself on fire.)
More can be said, but these answers from a year ago by u/PapiriusCursor and u/jlsturgeon may be helpful. reddit.com/r/AskHistorians/comments/56ty1f/were_the_picts_celtic_or_nonindoeuropean_people/ reddit.com/r/AskHistorians/comments/48tili/why_did_the_picts_become_gaelicized_rather_than/