The azimuthal anisotropy Fourier coefficients (${v}_{n}$) in 8.16 TeV $p+\mathrm{Pb}$ data are extracted via long-range two-particle correlations as a function of the event multiplicity and compared to corresponding results in $pp$ and PbPb collisions. Using a four-particle cumulant technique, ${v}_{n}$ correlations are measured for the first time in $pp$ and $p+\mathrm{Pb}$ collisions. The ${v}_{2}$ and ${v}_{4}$ coefficients are found to be positively correlated in all collision systems. For high-multiplicity $p+\mathrm{Pb}$ collisions, an anticorrelation of ${v}_{2}$ and ${v}_{3}$ is observed, with a similar correlation strength as in PbPb data at the same multiplicity. The new correlation results strengthen the case for a common origin of the collectivity seen in $p+\mathrm{Pb}$ and PbPb collisions in the measured multiplicity range.