You can simulate the Maple syntax quite easily:

f1 = x1^2 + x2^2;
f2 = x1*x2;
jac = D[{f1, f2}, {{x1, x2}}];
MatrixForm[jac]

The only notable difference is the display of the matrix.

Resource functions are just useful functions that have been submitted to the resource function repository by users. There is a whole system for submitting them and getting them accepted. They aren't part of the core language, so to access them, you need to go through the resource function repository. But again, there's D, which is core to the language, so I'm not sure what the problem is. The Jacobian is even mentioned in the documentation for D.

Side note: comparing Mathematica to Maxima or Maple is not really appropriate for this forum. This forum is just here to help folks with Mathematica. If you're suggesting feature requests or criticisms, that's something you should send to Wolfram directly, not this forum.

I don't know where you came across the symbol JacobianMatrix, but I don't think that's a built in symbol. You can just use the normal derivative.

D[{x1^2 + x2^2, x1 x2}, {{x1, x2}}]
(* {{2 x1, 2 x2}, {x2, x1}} *)

And just let me say proactively, that MatrixForm is only for display (which is fine for what you have so far), so don't try to compute further with that specific form.

Why do you keep ignoring D[{f1, f2}, {{x1, x2}}], which a couple of others have mentioned? (Assumes f1 = x1^2 + x2^2; f2 = x1 x2.)

Is it just because it's not named "Jacobian"? It's one of the problems when we adopt a person's name as the basis for calling a common object, such as the derivative, something else. (Of course, I believe the "Jacobian matrix" came into use before a vector-matrix-tensor view of multivariate functions and their derivatives became dominant. So there's a historical reason why "Jacobian" is used.)

Anyway, D[{f1, f2}, {{x1, x2}}] is shorter than Jacobian[{f1, f2}, {x1, x2}]. And so is Grad[{f1, f2}, {x1, x2}], which also works.

D[{f1, f2}, {{x1, x2}}] is the currently documented way to calculate the Jacobian (in the docs for D[]). The method from the version 2 Mathematica book is Outer[D, {f1, f2}, {x1, x2}], which also appears in the docs for D[]. When I search the help center for "jacobian" (lower case), D[] is the fourth hit in the desktop documentation; it's a disappointing eighth in the online docs, after the despisèd "JacobianMatrix" resource function.

You mentioned large sets of equations. It's not exactly clear what you mean, but here's what I do when I have equations and variables that I don't want retype. You need a list of the variables, and their order matters usually. If you can name them such that Sort[] puts them in the desired order, then great, you can just extract the variables from the formulas and Union[] them. Otherwise, you'll probably have to type them by hand.

vars = {x1, x2, x3, x4};
funcs ={f1, f2, f3, f4}; (* pretend they've been defined in terms of vars *)
jac = D[funcs, {vars}]