Radio jets are present in a diverse sample of AGN. However, the mechanisms of jet powering are not fully understood, and it remains unclear to what extent they obey mass-invariant scaling relations similar to those found for the triggering and fuelling of X-ray-selected AGN. We use the multi-wavelength data in the eFEDS field observed by eROSITA/Spectrum-Roentgen-Gamma (SRG) and LOFAR to study the incidence of X-ray and radio AGN as a function of several stellar mass (M)-normalised AGN power indicators. From the LOFAR-eFEDS survey, we defined a new sample of radio AGN, with optical counterparts from Legacy Survey DR9, according to a radio-excess relative to their host star formation rate. We further divided the sample into compact and complex radio morphologies. In this work, we used the subset matching to the well-characterised, highly complete spectroscopic GAMA09 galaxies (0<z<0.4). We release this value-added LOFAR-eFEDS catalogue. We calculated the fraction of GAMA09 galaxies hosting radio, X-ray, and both radio and X-ray AGN as functions of the specific black hole kinetic (lambda_Jet_) and radiative (lambda_Edd_) power. Despite the soft-X-ray eROSITA-selected sample, the incidence of X-ray AGN as a function of lambda_Edd_ shows the same mass-invariance and power law slope (-0.65) as that found in previous studies once corrected for completeness. Across the M range probed, the incidence of compact radio AGN as a function of lambda_Jet_ is described by a power law with constant slope, showing that it is not only high mass galaxies hosting high power jets and vice versa. This slope is steeper than that of the X-ray incidence, which has a value of around ~1.5. Furthermore, higher-mass galaxies are more likely to host radio AGN across the lambda_Jet_ range, indicating some residual mass dependence of jet powering. Upon adding complex radio morphologies, including 34 FRIIs, three of which are giant radio galaxies, the incidence not only shows a larger mass dependence but also a jet power dependence, being clearly boosted at high lambda_Jet_ values. Importantly, the latter effect cannot be explained by such radio AGN residing in more dense environments (or more massive dark matter haloes). The similarity in the incidence of quiescent and star-forming radio AGN reveals that radio AGN are not only found in "red and dead" galaxies. Overall, our incidence analysis reveals some fundamental statistical properties of radio AGN samples, but highlights open questions regarding the use of a single radio luminosity-jet power conversion. We explore how different mass and accretion rate dependencies of the incidence can explain the observed results for varying disk-jet coupling models.

Cone search capability for table J/A+A/686/A43/catalog (LOFAR-eFEDS value-added catalogue)