In this case you have categorial data. Maybe this could help you: https://towardsdatascience.com/hierarchical-clustering-on-categorical-data-in-r-a27e578f2995
Yes, you can use binary/dichotomous variables as the replications dimension for clustering cases. Of course, there will be a lot of tied scores within the data set, so you'd probably need a fair number of variables to develop any meaningful differentiation of groups/clusters. As an example, with two variables, the most groups/clusters you could have would be 4 (1,0 ; 1,1; 0,1; 0,0).
Would it perhaps make sense to combine some of the questions into subscales/subscores, rather than treat each survey question as an individual variable?
The linked IBM page is the right source to get info on this issue. SPSS two-step cluster analysis uses hierarchy in the clustering process, but in a way that allows the use of binary data as well as combining it with other types of data. It is a very flexible procedure.