Context. Several new ultrahigh-energy (UHE) γ-ray sources have recently been discovered by the Large High Altitude Air Shower Observatory (LHAASO) collaboration. These represent a step forward in the search for the so-called Galactic PeVatrons, the enigmatic sources of the Galactic cosmic rays up to PeV energies. However, it has been shown that multi-TeV γ-ray emission does not necessarily prove the existence of a hadronic accelerator in the source; indeed this emission could also be explained as inverse Compton scattering from electrons in a radiation-dominated environment. A clear distinction between the two major emission mechanisms would only be made possible by taking into account multi-wavelength data and detailed morphology of the source. Aims. We aim to understand the nature of the unidentified source LHAASO J2108+5157, which is one of the few known UHE sources with no very high-energy (VHE) counterpart. Methods. We observed LHAASO J2108+5157 in the X-ray band with XMM-Newton in 2021 for a total of 3.8 hours and at TeV energies with the Large-Sized Telescope prototype (LST-1), yielding 49 hours of good-quality data. In addition, we analyzed 12 years of Fermi-LAT data, to better constrain emission of its high-energy (HE) counterpart 4FGL J2108.0+5155. We used naima and jetset software packages to examine the leptonic and hadronic scenario of the multi-wavelength emission of the source. Results. We found an excess (3.7σ) in the LST-1 data at energies E > 3 TeV. Further analysis of the whole LST-1 energy range, assuming a point-like source, resulted in a hint (2.2σ) of hard emission, which can be described with a single power law with a photon index of Γ = 1.6 ± 0.2 the range of 0.3 − 100 TeV. We did not find any significant extended emission that could be related to a supernova remnant (SNR) or pulsar wind nebula (PWN) in the XMM-Newton data, which puts strong constraints on possible synchrotron emission of relativistic electrons. We revealed a new potential hard source in Fermi-LAT data with a significance of 4σ and a photon index of Γ = 1.9 ± 0.2, which is not spatially correlated with LHAASO J2108+5157, but including it in the source model we were able to improve spectral representation of the HE counterpart 4FGL J2108.0+5155. Conclusions. The LST-1 and LHAASO observations can be explained as inverse Compton-dominated leptonic emission of relativistic electrons with a cutoff energy of 100−30+70 TeV. The low magnetic field in the source imposed by the X-ray upper limits on synchrotron emission is compatible with a hypothesis of a PWN or a TeV halo. Furthermore, the spectral properties of the HE counterpart are consistent with a Geminga-like pulsar, which would be able to power the VHE-UHE emission. Nevertheless, the lack of a pulsar in the neighborhood of the UHE source is a challenge to the PWN/TeV-halo scenario. The UHE γ rays can also be explained as π0 decay-dominated hadronic emission due to interaction of relativistic protons with one of the two known molecular clouds in the direction of the source. Indeed, the hard spectrum in the LST-1 band is compatible with protons escaping a shock around a middle-aged SNR because of their high low-energy cut-off, but the origin of the HE γ-ray emission remains an open question.