The reaction between hydrazine and HCl is very simple and completeluy analogous to the reaction between ammonia and HCl. In a solution of HCl you have H(+) and Cl(-) ions. The Cl(-) ions do nothing and are spectator ions.
With ammonia you have:
NH3(aq) + H+ ---> NH4+(aq)
On evaporation you get ammonium chloride NH4Cl.
Completely analogous is the reaction with hydrazine:
N2H4(aq) + H+ ---> N2H5+(aq)
On evaporation you get hydrazinium chloride N2H5Cl (this also is called hydrazine monohydrochloride and written as N2H4.HCl, but the latter notation is not really good).
In excess acid, the reaction even goes further on the other NH2-group of hydrazine:
N2H5+(aq) + H+ ---> N2H62+(aq)
On evaporation you get N2H6Cl2 (this is called hydrazine dihydrochloride and written as N2H4.2HCl, but the latter notation is not really good).
The latter salt is commercially available (in fact, I have some in my home lab and it is a white crystalline solid, which is quite acidic, when dissolved in water).