That's not quite right. TM's in general are not universal. The "universality" of TMs has to do with the existence of a universal TM which is capable of emulating any other TM by putting a specification of the TM to be emulated on the universal TM's tape. The reason this matters is that once you've built a universal TM, you never have to build any more hardware. Any TM can then be emulated in software. That result is due to Turing.
The relationship between TMs and mathematical logic is of a fundamentally different character. It turns out that any system of formal logic can be emulated by a TM (and hence by a UTM), but that is a different result, mainly due to Godel, not Turing. There is also the empirical observation (famously noted by Eugene Wigner [1]) that all known physical phenomena (with the exception of quantum measurements) can be modeled mathematically, and hence can be emulated by a TM, and hence can be emulated by a UTM. But it is entirely possible that a new class of physical phenomena could be discovered tomorrow that cannot be modeled mathematically.
But here's the thing: no one has been able to come up with any idea of what such a phenomenon could possibly look like, and there is a reason to believe that this is not just a failure of imagination but actually a fundamental truth about the universe because our brains are physical systems which themselves can be modeled by mathematics, and that is (empirical) evidence that our universe is in some sense "closed" under Turing-equivalence (with quantum randomness being the lone notable exception). That is the kind of universality embodied in the idea that TM's can emulated "anything we can dream up". It's called the Church-Turing thesis [2] and unlike the universality of universal TM's it cannot be proven because a counterexample might be discovered at any time.
[1] https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...
[2] https://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis