It depends what you define as chip design.
Coding vhdl or verilog is not chip design. It is system design.
Designing the actual and, or , not , doing the layout , extract parasitics and make a library of standard cells is chip design. You need to know how to create a transistor, how to define the wells, where to put the bulk connection, isolation rings and do it all in such a fashion that it yields the smallest surface area, fastest speed and lowest powerconsumption without blowing up in your face. The circuit needs to work reliably over process variations (layer thickness etc ) and temperature span.
For analog there is even more involved. You cannot simply say : i want 1 kiloohm here.. You'll get something that will be between 800 ohm and 4k...
Only the ratios will work. Absolute values are unknown. Cant make capacitors either on chip except for extremely small ones... So everything is dc coupled in your design....
There is a linrary with standard blocks but most commonly these are taken, modified a bit and then used. You cant simply slap down 4 opamps and wire them up with 5 resistors.... The design would be waaay too large. You'd strip the output stage for the opamps used internally. Youll also strip,the bias generators as you can share the bias generator between opamps.
So that stuff invariably ends up being full custom starting from a few base blocks with a full custom layout. You will need to interact with the ic layouters as they will extract parasitics and you will have to see if there is impact and provide feedback. Select carefully as any square micron you add increase cost of the chip thus reducing profit....
What block to place where ? How do we bond-out. How do we thermally distribute ? What about gradient compensation ?
There is a lot of factors to account for that do not exist in traditional design.
Then there is the whole testability problem. You will need to inject test multiplexers to be able to probe internal nodes during ic testing. Every singgle chip produced goes through that.
A circuit you breadboard with an lm324 and some resistors in half an hour takes about a week to design on silicon... Design as in : here is the GDS tape , please make the masks and fab it.