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 ($ Jet $) and radiative ($ Edd $) power. Despite the soft-X-ray eROSITA-selected sample, the incidence of X-ray AGN as a function of $ 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 $ 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 $ 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 $ 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.