You are writing "62.1" but you are trying to read up until a
\n, which you never send.
You need to ensure that you send the end-of-line character that you subsequently check for,
Also it's worth noting that floating point values are only ever an approximation. They're great for doing calculations, but very poor at doing comparisons. Your test value of 62.1 cannot be represented precisely as a floating point value, the closest being about 62.099. If you do need to do a comparison then you should check it's within a small range, not a precise value.
Your "problem" seems to stem from you opening the serial port with one program, sending a value, then closing the serial port and opening it again in something else and expecting to see that value being echoed back to you.
That can't happen on a stock Arduino UNO. Every time you open the serial port the DTR line toggles the RESET pin to enter the bootloader - which is why you need the 2 second delay after opening the serial port.
In doing so the sketch then runs fresh every time you open the serial port.
There are a number of ways of disabling that reset, from cutting a trace on the Arduino, to adding a large-ish (10µF +) capacitor between RESET and GND - both of which will interfere with your ability to program the board. The capacitor is the easier to reverse.
Ideally you'd use a different serial channel to communicate so it never interferes with the reset or programming. You could use
SoftwareSerial and an external USB to UART adaptor instead of the
Serial connection, or use a board with a native USB connection that doesn't reset when you open the serial port, such as a Leonardo.
abs(val - 62.1) < 0.05where 0.05 is the 'accuracy you accept, so values between 62.05 and 62.15 are resulting in