Literally the same Deity. Rudra (Shiva) & Odin.
To this, if we were really going to get into it, we would also be adding an array of other stone or wood focal points for worship of the Sky Father deific, found throughout the Indo-European sphere.
Indeed, not just ‘focal points’ in the sense of altar-stones or altarpieces where offerings would be carried out – often actively understood to be ‘forms’ or ‘presences’ of the God Himself.
This is partially why we find examples like the fine Gupta-era Mukhalingam (‘Face-Lingam’) in the top panel of this image (others will have multiple ‘faces’, one for each cardinal direction, presenting particular Aspects to the God – something we might be tempted to speculate may resonate with certain of the polycephalic or otherwise multi-faced Slavic deific representations)
And yes, before somebody asks – you can, in fact, have wooden ShivLings. These are referred to as Daruja-Lingas (literally Wooden Lingas). Nevertheless, we most usually think of the large riverstone style ones, so these (and the other ways one can make a ShivLing detailed in the scripture) get much less attention.
Of course, we should probably note that when the Iuppiter Lapis [‘Jupiter Stone’] is referred to in academic literature as being a “rude stone” … this has, perhaps, a different meaning to what some might think ‘rude stone’ meant when looking upon some of the more overt styles of ShivLing presentation.
Instead, “rude stone” there refers to the quality of it being unworked by mortal hands [as compared to, say, the decidedly carved renderings of the Jupiter Columns found in Germania]. What we would term for one of our ShivLings as the quality of Swayambhu [‘Self-Manifested’].
However, as applies the other (and more modern) meaning for ‘rude stone’ … in addition to the obvious circumstance of the Herms of Greece, we would also draw attention to the situation reported by St Augustine (apparently drawing from the work of Varro), viz. Liber (i.e. Bacchus / Dionysus) –
“Among other rites which I am compelled to pass over due to their sheer number, there are certain rites of Liber that [Varro] says are celebrated at Italian crossroads with such shameful abandon that the private parts of the male are worshipped in the god’s honour, and not even in secret, out of some deference to modesty, but openly and with wantonness running riot. Yes indeed, during Liber’s festal days this disgusting member would be exhibited on a waggon with great honour, and carried first around the crossroads in the countryside, and then brought all the way into the city. In the town of Lavinium they even dedicated a whole month to Liber, and during these days they all used such disgraceful language until that member had been carried through the forum and come to rest in its own place. Then the most respectable woman, a mother of a family, had to place a garland on said dishonourable member. In this way, supposedly, the god Liber was to be propitiated so that all would turn out well with the seeds; in this way the hex needed to be warded off the fields…” ( City of God 7.21.2–4; Varro ARD 262  Cardauns)
To take things further, in terms of active worship:
In the Gylfaginning, we find the following:
“It is further said that these Norns who dwell by the Well of Urdr take water of the well every day, and with it that clay which lies about the well, and sprinkle it over the Ash, to the end that its limbs shall not wither nor rot; for that water is so holy that all things which come there into the well become as white as the film which lies within the egg-shell,–as is here said:
I know an Ash standing | called Yggdrasill,
A high tree sprinkled | with snow-white clay;
Thence come the dews | in the dale that fall–
It stands ever green | above Urdr’s Well.
That dew which falls from it onto the earth is called by men honey-dew, and thereon are bees nourished. Two fowls are fed in Urdr’s Well: they are called Swans, and from those fowls has come the race of birds which is so called.”
We have written voluminously about the cognate broader Indo-European resonancies for what has just been attested there, and shall not repeat it here, except to note that:
i) this phenomenon is “something that is also known via the curious Greek ‘double-meanings’ for the Ash Tree – as ‘Meliai’ (not at all coincidentally a term that informs words linked to ‘Spear’), but also producing ‘Meli’ (Ancient Greek for Honey), and the Meliai (singular – ‘Melia’), the Ash-Nymphs that look after the infant Zeus and feed Him Honey (Meli again) and the Milk of Amalthea.”
[to quote myself upon the subject]
ii) the word being utilized to translate as ‘clay’ – ‘auri’ – isn’t exactly ‘clay’, but rather connotes a wet substance (in fact, given the white liquid of the relevant Wellspring of Urðr is the key ingredient, the Bellows translation not unreasonably has ‘Water’ in lieu of ‘Clay’ there for the libation); and to my mind, recalls our smearing of the ShivLing with light-coloured ritual unguents – or, for that matter, the rather thicker nature of a Panchamrut [Panchamrit – Five-Element Amrit] libation due to the yoghurt / curd etc. involved (along with the Honey, jaggery (raw cane sugar), ghee, and of course, the milk … a pointedly white liquid of life).
iii) that the situation outlined above of a daily oblation to the World-Tree / Axis Mundi, utilizing something quite recognizable to our ShivLing conceptry – would quite likely inform how the Irminsul was worshipfully engaged with by the Germanics.
After all, per Ruodolf of Fuld’s description, they hailed it as being the “universalis columna, quasi sustinens omnia” – the ‘pillar of the universe which supports all’.
Just as, funnily enough, we regard the ‘ultimate’ Lingam from whence all of our ShivLings resonantly descend / emanate in principle.
There are some further points we could (and almost unquestionably should) make viz. Irmin in relation to Odin, and why we adhere to that particular identification … but for another time.
iv) that notion briefly aforementioned viz. an ‘infant’ form of the Sky Father being given vital milk finds intriguing resonancy when we consider the Omphalos stone of Delphi (a ‘center of the world’) … and its alleged situation as a ‘doublet’ for the Infant Zeus in some tellings of the relevant myth. But I digress.
One final point, and again quoting myself from earlier work:
“Of further interest for our purposes is an intriguing set of references in both Pseudo-Apollodorus’ Bibliotheca and Strabo’s Geography to Dionysus in India in relation to ‘Pillars’. In the former case, wandering India personally setting up these structures; in the latter, said ‘pillars’ only being found where the “Dionysus” as the Macedonians knew Him was venerated (or, to be sure, where Herakles was hailed, also).”
The situation viz. Herakles would, on the surface, seem a troublesome one to reconcile – however, we are well-familiar with the pole- / post-raising rite for Indra … that is, as it should happen, still celebrated in Nepal today.
Of course, the main Vedic religious element we must invoke here is the Sthambha … which, unsurprisingly, is significantly Shaivite in both its subsequent ’emanations’ or ‘descendants’ (those being the ShivLings capaciously aforementioned) , as well as the most prominent appearance of it in the Four Vedas – where it is not ‘just’ a Sacral Post in the general sense, but rather the Post which supports and binds together the Worlds Entire.
In AV-S X 7 and 8 we find it extolled – and hailed elsewhere as the Jwala-Linga. An appropriately Infinite form for Lord Shiva.