But #include <sycl/sycl.hpp> should be enough to compile the application, if it is claimed to be standard C++. So, is it possible to do?

Yes. As you noted in the comments, SimSYCL is an example of a library-only implementation that does not require any compiler support (although, it requires C++20 standard and is not production-oriented).

More practically, AdaptiveCpp has library-only mode which can target CPUs with any compiler supporting OpenMP, and NVIDIA GPUs with NVIDIA NVC++ compiler.

I have not found any evidence that it is possible, so should I understand that SYCL source code only looks like standard C++?

Language-lawyering a bit, but C++ standard is exactly about what code "looks like". And I would not underestimate the importance of it. E.g., being standard compliant, SYCL code does not require any special handling from code highlighters, formatters, linters, static analyzers etc (unless we're talking about device-specific analysis).