Some audio operations may require greater precision, such as fft.
I have tested a float32 fast fourier transform, and transform the signal back, and the error is about -133dB (maybe the number is wrong, anyway, is worse than -144dB)

I am thinking that some other complex audio operations might be similar. I shudder to think of the cumulative effect if there were many such operations. Hopefully double will be supported, after all the export screen still has the option to double.
By the way, can the internal precision of the plugins be controlled by the DAW?

Not quite. First keep in mind that thermal noise-floor at room-temperature is at about -127dBm (@ 48kHz bandwidth). As interface to/from the analog world you get at best 18-19 bits of valid information. Also in between processors any potential rounding issues are negligible:

The smallest possible value that is added/removed when rounding with a 23 bit mantissa and 1 sign-bit, fixed exponent at 1.0 is 2^-23. Worst case, each arithmetic float operation will add an error of 2^-23.

The noise added due to rounding errors after N worst-case operations is
20 * log10 (N * (2^-23)) dB.

In reality with random, not-additive errors accumulating, millions of float operation can be safely performed without introducing significant noise above quantization level.

Inside specific processors it can make sense to use higher precision for coefficients and in some cases also higher sample-rates, but as interface between processors it is useless (unless you’re in marketing and want to sell new hardware).

In your case the FFT inside a plugin may use double precision even though the input and output is 32bit floating point. When using cookbook biquad filters the coefficients for high-pass or low-frequency filters are close to 1.0 where precision of 32bit float is not very good. Internally some plugins use double-precision float for coefficients for that reason. A good resource here is the article What Every Computer Scientist Should Know About Floating-Point Arithmetic. Some distortion effects oversample internally to reduce aliasing.

In any case, just blindly using double and/or higher sample-rate does not solve anything.

–

PS. see also: The Scientist and Engineer’s Guide to Digital Signal Processing
By Steven W. Smith, Ph.D. Chapter 4 - Number Precision

No. The host has no control of what a plugin does.

This is also an argument why a host should not offer a bypass for plugin(s). Only the plugin knows how to click-free bypass itself. Most plugins have dedicated bypass controls. e.g. an EQ should internally ramp gain per band to zero.