The azimuthal anisotropy Fourier coefficients $(vn)$ in $8.16 TeV p+Pb$ 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 $p+Pb$ collisions. The$ v2$ and $v4$ coefficients are found to be positively correlated in all collision systems. For high-multiplicity $p+Pb$ collisions, an anticorrelation of $v2$ and $v3$ 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 $p+Pb$ and $PbPb$ collisions in the measured multiplicity range.