TensorFlow - why doesn't this sofmax regression learn anything?

I hope this helps you . The main problem is that you are feeding just one sample at a time to the model. This makes your optimizer very inestable, that's why you have to use such a small learning rate. I will suggest you to feed more samples in each step.
If you insist in feeding one sample at a time, maybe you should consider using an optimizer with momentum, like tf.train.AdamOptimizer(learning_rate). This way you can increase the learning rate and reach convergence.

seems to work fine The problem turned out to be silly, I just needed more epochs, a smaller learning rate (and for efficiency I turned to AdamOptimizer, results are now equal, although the TF implementation is much slower.

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SCI-KITLEARN RESULTS:
    Accuracy: 0.822129684711
    Precision: 0.837883361162
    Recall: 0.784522522208
    F1: 0.806251963817

TENSORFLOW RESULTS:
    Accuracy: 0.82213
    Precision: 0.837883361162
    Recall: 0.784522522208
    F1: 0.806251963817

wish help you to fix your issue I just compare the results by tensorflow and numpy. Since you used dtype=tf.float32 for X and y, I will use np.float32 for the numpy example as follows:

X_numpy = housing_data_plus_bias.astype(np.float32)
y_numpy = housing.target.reshape(-1, 1).astype(np.float32)

with tf.Session() as sess:
    XTX_value = tf.matmul(XT, X).eval()
XTX_numpy = X_numpy.T.dot(X_numpy)

np.allclose(XTX_value, XTX_numpy, rtol=1e-06) # True
np.allclose(XTX_value, XTX_numpy, rtol=1e-07) # False

This might help you If the only thing you are doing is regression, scikit-learn is good enough and will definitely do you job. Tensorflow is more a deep learning framework for building deep neural networks.
There're people using Tensorflow to do regression maybe just out of personal interests or they think Tensorflow is more famous or "advanced".

hop of those help? I tried to run your examples and noticed a couple of potential sources:
The test set is incredibly small, only 45 instances. This means that to get from accuracy of .89 to .96, the model only needs to predict just three more instances correctly. Due to randomness in training, your Keras results can oscillate quite a bit. As explained by @meowongac https://stackoverflow.com/a/59643522/1467943, you're using a different optimizer. One point is that scikit's algorithm will automatically set its learning rate. For SGD in Keras, tweaking learning rate and/or number of epochs could lead to improvements. Scikit learn quietly uses L2 regularization by default.