I think that what you describe is a standard application of multivariate functional data clustering. In the context of multivariate functional data each data unit is treated as the relation of a $d$-dimensional stochastic (often Gaussian) process $X := ( X_1, \dots , X_d )$.
Jacques & Preda (the authors of the nice survey paper you attach) have (somewhat) recently published a paper on "Model-based clustering for multivariate functional data (2014)" which extends their earlier work on "Clustering multivariate functional data (2012)". Approximately at the same time Chiou et al. also on "Multivariate functional principal component analysis: A normalization approach (2014)". Note that the two approach are quite different; Chiou's approach has a particular (very flexible) parametric association between the curve-samples while Jacques & Preda is much more data-driven.
Both of these works are based on multivariate functional principal component analysis (MvFPCA). Earlier applications where alluded in Ramsay & Silverman's "Functional Data Analysis (2005)" book when they looked at hip and knee angle curves at the same time but they did not follow up. A underappreciated work is actually Yang's et al. work on "Functional singular component analysis (2011)", they seems to have somewhat stumbled upon MvFPCA but then decided to focus on the analysis of the singular values (in fairness they deal only with two-processes' data). Effectively Berrendero et al. with their work on "Principal components for multivariate functional data (2011)" gave the first full treatment of the subject.
MvFPCA by itself is still an active domain of research; getting the good basis in higher dimensions is a tricky part, many non-parametric approaches like spline and kernel smoothers get really variable. Happ & Greven have 2015 arXiv paper on the matter of "Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains"; in that work for example for 3-dimensional data they look into DCT as the basis-of-choice (so something fully parametric) and leave non-parametric basis estimation to the side. (This paper is now published in JASA)
Having said all that, if you look at the Jacques & Preda (2014) you will see that given you project the data to a lower dimensional manifold, multivariate clustering techniques are rather competitive. As a first, easy pass I would suggest you do just that: use an MvFPCA approach of your choice, get the projection scores and try a standard well-understood multivariate clustering approach. I have found mixture-models (like the ones in the CRAN package
EMCluster) to work reasonably well but I am sure other techniques can give fruitful results too.
Software-wise in R, the CRAN package
Funclustering appears to have some MvFPCA out of the box; Happ & Greven (2015) have a package named
MFPCA on github, Yang's et al. Singular Decomposition is available in the CRAN package