This depends on how you want to set up your system.
A TestFunction and a TrialFunction are both arguments, instructing UFL what kind of object that you would like to assemble.

If you have no test or trial function in a variational form, the resulting assembled object will be a scalar value.

If you have either a test or a trial function in your variational form, the assembled result will be a vector.

If you have both a test and trial function in your variational form, the result will be a matrix.

If you solve a non-linear problem, it is common to define the residual of the variational form F(u,v)=0 and then differentiate it to get the linearized Jacobian (for instance described at: Custom Newton solvers — FEniCSx tutorial). Then you would use a dolfinx.fem.Function, and not a ufl.TrialFunction to define the “unknown” in your form.