`FunctionTransformer` need `feature_names_out` even if `func` returns DataFrame

### Describe the bug

Trying to call `transform` for `FunctionTransformer` for which `feature_names_out` is configured raises error that advises to use `set_output(transform='pandas')`. But this doesn't change anything.

### Steps/Code to Reproduce

```py
import numpy as np
import pandas as pd
from sklearn.preprocessing import FunctionTransformer

my_transformer = FunctionTransformer(
    lambda X : pd.concat(
        [
            X[col].rename(f"{col} {str(power)}")**power
            for col in X
            for power in range(2,4)
        ],
        axis=1
    ),
    feature_names_out = (
        lambda transformer, input_features: [
            f"{feature} {power_str}"
            for feature in input_features
            for power_str in ["square", "cubic"]
        ]
    )
)
# I specified transform=pandas
my_transformer.set_output(transform='pandas')
sample_size = 10
X = pd.DataFrame({
    "feature 1" : [1,2,3,4,5],
    "feature 2" : [3,4,5,6,7]
})
my_transformer.fit(X)
my_transformer.transform(X)
```

### Expected Results

`pandas.DataFrame` like following

|    |   feature 1 square |   feature 1 cubic |   feature 2 square |   feature 2 cubic |
|---:|-------------------:|------------------:|-------------------:|------------------:|
|  0 |                  1 |                 1 |                  9 |                27 |
|  1 |                  4 |                 8 |                 16 |                64 |
|  2 |                  9 |                27 |                 25 |               125 |
|  3 |                 16 |                84 |                 36 |               216 |
|  4 |                 25 |               125 |                 49 |               343 |

### Actual Results

```
ValueError: The output generated by `func` have different column names than the ones provided by `get_feature_names_out`. Got output with columns names: ['feature 1 2', 'feature 1 3', 'feature 2 2', 'feature 2 3'] and `get_feature_names_out` returned: ['feature 1 square', 'feature 1 cubic', 'feature 2 square', 'feature 2 cubic']. The column names can be overridden by setting `set_output(transform='pandas')` or `set_output(transform='polars')` such that the column names are set to the names provided by `get_feature_names_out`.
```

### Versions

```shell
System:
    python: 3.10.12 (main, Nov 20 2023, 15:14:05) [GCC 11.4.0]
executable: /usr/bin/python3
   machine: Linux-6.5.0-14-generic-x86_64-with-glibc2.35

Python dependencies:
      sklearn: 1.4.1.post1
          pip: 24.0
   setuptools: 68.2.2
        numpy: 1.24.2
        scipy: 1.11.1
       Cython: None
       pandas: 2.2.1
   matplotlib: 3.7.1
       joblib: 1.3.1
threadpoolctl: 3.1.0

Built with OpenMP: True

threadpoolctl info:
       user_api: blas
   internal_api: openblas
         prefix: libopenblas
       filepath: /home/fedor/.local/lib/python3.10/site-packages/numpy.libs/libopenblas64_p-r0-15028c96.3.21.so
        version: 0.3.21
threading_layer: pthreads
   architecture: Haswell
    num_threads: 12

       user_api: openmp
   internal_api: openmp
         prefix: libgomp
       filepath: /home/fedor/.local/lib/python3.10/site-packages/scikit_learn.libs/libgomp-a34b3233.so.1.0.0
        version: None
    num_threads: 12

       user_api: blas
   internal_api: openblas
         prefix: libopenblas
       filepath: /home/fedor/.local/lib/python3.10/site-packages/scipy.libs/libopenblasp-r0-23e5df77.3.21.dev.so
        version: 0.3.21.dev
threading_layer: pthreads
   architecture: Haswell
    num_threads: 12
```
```

COMMENT:
> The error message might miss an information: if you use set_output then, you can remove feature_names_out

I was expecting something similar but it doesn't seem to work as I hinted above and the snippet in https://github.com/scikit-learn/scikit-learn/issues/28780#issuecomment-2045790268 shows. You need to specify `feature_names_out` otherwise `.get_feature_names_out` does not exist ...

Labels: Bug