The question:
Is the murre, Uria (Fig. 1), more closely related to penguins, or to auks?

The answer is BOTH are correct – if you add one more taxon: another ‘Uria‘.
Uria aalge, the extant common murre (pronounced ‘myur’), is basal to penguins in the large reptile tree (LRT, 1897+ taxa, subset Fig. 1), as it has been for several years. Newly added Uria lomvia, the thick-billed murre, is basal to auks. Who saw this generic split coming? Not me. Evidently no one.

These two taxa look alike and they live together with identical lifestyles,
but “Don’t Pull a Larry Martin!” This is another overlooked example of the rule against listing similar traits to support a presumed relationship. Instead: Be a scientist. Don’t assume. Add taxa. Run analysis. Find out where your taxa nest. Don’t rely on a handful of traits, sizes, niches or color schemes. Don’t rely on textbooks or traditions. Find out for yourself. If I can do it, others can, too. Even paid professionals at universities. Like everything in science, this hypothesis needs to be confirmed over and over again. Or refuted. Don’t wait for another twenty years to pass.

Think of these two murres like seals and sea lions,
or like odontocetes (toothed whales) and mysticetes (baleen whales). Superficially they may look alike, but on a phylogenetic level, they are not alike. In the LRT many other taxa are more like these two than they are to each other (Fig. 1). So far (apparently due to taxon exclusion), this excellent example of convergence has escaped the notice of taxonomists. (Let me know if this is an oversight).

The first named of these murres,
Uria lomvia (Linneaus 1758), remains a murre with this generic name. After analysis the later named, Uria aalge (Pontoppidan 1763), should be known henceforth as a murre-mimic (Fig. 1) and it needs a new generic name.

Once again,
adding taxa resolves phylogenetic problems. Don’t let another future example of this low-hanging fruit get picked by retired maligned armchair amateurs sitting at their monitors.

This new insight started with a deep dive into Watanabe et al. 2021
who studied penguins (Fig. 2) and auks (Fig. 3) to understand how these unrelated wing-propelled underwater birds evolved convergently, radiating into two different hemispheres, the north for auks and the south for penguins.

According to Watanabe et al.
(who borrowed their cladogram (Fig. 4) from the gene-based study by Prum et al. 2015):

1. Penguins (like Spheniscus and Aptenodytes, Fig. 2) are derived from the long-winged procellariformes (tube-nosed soaring birds like Calonectris Fig. 4) even though no penguins have long wings or a tube nose. By contrast, in the LRT (subset Fig. 1) Calonectris is closer to Catharacta and then auks.
2. Auks (like Pinguinis and Alca, Fig. 3) are derived from murres (like: Uria lomvia). Uria aalge was not tested by Watanabe et al, but it nests closer to penguins in the LRT.

Prum et al. 2015,
as you might remember, created a cladogram of mismatched taxa (e.g. flamingos close to grebes) based on genes. Watanabe et al. accepted that analysis without testing it. This is common and accepted practice in vertebrate paleontology. Sometimes it works, but not all the time. So don’t take the chance.

By contrast
in the LRT (subset Fig. 1), which tests traits and fossils, auks arise from murres, and so do penguins. Auks arise from Uria lomvia (Fig. 3). Penguins arise from Uria aalge (Fig. 2), here considered a misnamed genus due to overlooked convergence and taxon exclusion in prior descriptions and phylogenetic studies.

Since all murres inhabit the same northern geographical areas,
it is possible that they will also nest together in genetic studies having incorporated the same geographic viruses. That’s the problem with deep time gene studies.

It is easy to see how murres
would be good ancestral candidates for both auks and penguins. Murres can fly in the air AND underwater (as deep as 300 feet) with their relatively short wings adapted to both environs.

In the LRT penguins and auks have their own set
of ancestors and relatives (Fig. 1) with no last common ancestor until close to long-legged plovers like Charadrius. The convergence between auks and penguins began much earlier, in their flying ancestors among the murres and murre-mimics.

In both gene and trait studies
the great auk (Pinguinis) nested with the razorbill (Alca, Fig. 3). Auk outgroups in both studies include the puffin (Fratercula). auklet (Cerorhinca) and murre (Uria lomvia). So here (Fig. 5) is an example of a gene study that match a trait study over several nodes. But don’t take that as a good sign. Remain wary. False positives are sprinkled with validated positives. For example:

Tube nosed birds (Procellariformes)
(Fig, 4) are a clade all on their own in the LRT, specialized for soaring and not giving rise to any short wing-swimming taxa. like penguins (Fig. 6). In gene studies Catharacta nests apart from Calonectris (Fig.4). In trait studies, like the LRT (Fig. 1), they nest closer together and apart from penguins.

Somehow Watanabe et al. added a fossil taxon,
Mancalla (Fig. 5 red line), into a gene study. Oddly this flightless auk nested basal to volant taxa, creating a third clade of flightless diving birds. Mancalla was not tested in the LRT. It needs to be tested based on traits with other taxa based on traits.

Moving one Uria over to the other, and vice versa,
in the LRT adds 30 and 34 steps to the most parsimonious tree. So the difference is substantial.

Bottom line:
The two tested Uria genera are not congeneric.

References
Linnaeus C 1758. Systema naturæ per regna tria naturæ, secundum classes, ordines, genera, species, cum characteribus, differentiis, synonymis, locis. Tomus I. Editio decima, reformata.
Pontoppidan E 1763. Den Danske atlas eller Konge-Riget Dannemark, med dets naturlige egenskaber, elementer, indbyggere, værter, dyr og andre affødninger, dets gamle tildragelser og nærværende omstændingheder i alle provintzer, stæder, kirker, slotte og herre-gaarde. Forestillet ved en udførlig lands-beskrivelse, saa og oplyst med dertil forfærdigede land-kort over enhver provintz, samt ziret med stædernes prospecter, grund-ridser, og andre merkværdige kaabber-stykker. Efter Søy-kongelig allernaadigst befalning. 1: 1-723.
Prum et al. (6 co-authors) 2015. A comprehensive phylogeny of birds (Aves) using targeted next-generation DNA sequencing. Nature 526:569–573. online
Watanabe J, Field DJ and Matsuoka H 2021. Wing Musculature Reconstruction in Extinct Flightless Auks (Pinguinus and Mancalla) Reveals Incomplete Convergence with Penguins (Spheniscidae) Due to Differing Ancestral States. Integrative Organismal Biology, a journal of the society of integrative and comparative biology pp. 1-64.

References
wiki/Great_auk
wiki/Razorbill
wiki/thick-billed murre – Uria lomvia
wiki/Aptenodytes
wiki/Uria
wiki/Common_murre
wiki/Galapagos_penguin