In summary, this is poor practice. While not "wrong" in an absolute sense it introduces avoidable risks.
SQL Server optimizes each statement separately. The execution plan for the "everything" stored procedure will not be noticeably different to the combined plans of having each bit in its own stored procedure (SP). There will be a very tiny overhead of evaluating the IFs, but you would never measure it, much less notice it.
It seems they're treating the SP as a plug-in library. SPs are not libraries. They should be treated as methods. Each will have its own clear, distinct purpose and implementation.
How would code review treat application code that had a class with a single mega-method? Would the devs accept this style in the UI or business tier? Likely not. Don't do it in the DB tier either.
There are some run-time implications. The mega-SP has a single plan. Changing any part of the code will cause the whole plan to be re-compiled, stalling all clients wishing to run any part of that plan, even if they're executing a function whose code has not changed. The time to compile a large SP is longer than the time to compile a small one. Plans have to be loaded in toto to be executed, even if only a small fraction of the code path is traversed. So fewer plans can be in memory at any one time, meaning plans get evicted from cache more often, causing re-comiles and client slowness. If the server has more memory than it needs and is never under memory pressure this may never manifest as an actual issue.
Project management becomes more complicated, too. With a single SP lots of changes will affect that one module risking code merge fails. (Your DB code is in source control, right?)
The code is less self-documenting. What does function "1" mean in this SP? Is it these same meaning in that SP? Should every SP have a function "1"?
The parameters must be a superset of all the functions. Now you need even more documentation to say "function 1 uses @a, @c and @f; function 2 uses @a and @g but don't populate @c or year-end crashes." Likely every parameter will be optional losing further built-in documentation and validation.