Analysis of $T_{cc}$ and $T_{bb}$ based on the hadronic molecular model and their spin multiplets

${T_{cc}(cc\bar{u}\bar{d})}^{+}$ has been reported by the LHCb experiment in 2022. The analysis using the Breit-Wigner parametrization found the small binding energy, $0.273$ MeV, which is measured from the threshold of $D^{*+}D^{0}$. In this paper, we consider $T_{cc}$ as a $DD^*$ hadronic molecule as a deuteron-like state. The one boson exchange model is employed as for the heavy meson interactions, where we determine the cut-off parameter $\Lambda$ to reproduce the reported binding energy of $T_{cc}$ with $I(J^P)=0(1^+)$. We discuss the properties of the bound state, and also search for $T_{cc}$ with the quantum numbers other than $0(1^{+})$. Futhermore, we analyze $T_{bb}$ as a bottom counterpart of $T_{cc}$, which involves two bottom quarks, and obtain several bound states. Finally we consider the light-cloud basis for wave functions of the doubly heavy tetraquarks in the heavy quark limit. Using the basis, we find the spin multiplets of their bound states, indicating the spin structures of diquarks in $T_{cc}$ and $T_{bb}$ with the finite quark masses.