How to merge keras sequential models with same input?

Keras functional API seems to be a better fit for your use case, as it allows more flexibility in the computation graph. e.g.:

from keras.layers import concatenate
from keras.models import Model
from keras.layers import Input, Merge
from keras.layers.core import Dense
from keras.layers.merge import concatenate

# a single input layer
inputs = Input(shape=(3,))

# model 1
x1 = Dense(3, activation='relu')(inputs)
x1 = Dense(2, activation='relu')(x1)
x1 = Dense(2, activation='tanh')(x1)

# model 2 
x2 = Dense(3, activation='linear')(inputs)
x2 = Dense(4, activation='tanh')(x2)
x2 = Dense(3, activation='tanh')(x2)

# merging models
x3 = concatenate([x1, x2])

# output layer
predictions = Dense(1, activation='sigmoid')(x3)

# generate a model from the layers above
model = Model(inputs=inputs, outputs=predictions)
model.compile(optimizer='adam',
              loss='binary_crossentropy',
              metrics=['accuracy'])

# Always a good idea to verify it looks as you expect it to 
# model.summary()

data = [[1,2,3], [1,1,3], [7,8,9], [5,8,10]]
labels = [0,0,1,1]

# The resulting model can be fit with a single input:
model.fit(data, labels, epochs=50)

Notes:

• There might be slight differences in the API between Keras versions (pre- and post- version 2)
• The example above specifies different optimizer and loss function for each of the models. However, since fit() is being called only once (on model3), the same settings - those of model3 - will apply to the entire model. In order to have different settings when training the sub-models, they will have to be fit() separately - see comment by @Daniel.

EDIT: updated notes based on comments

etov's answer is a great option.

But suppose you already have model1 and model2 ready and you don't want to change them, you can create the third model like this:

singleInput = Input((3,))

out1 = model1(singleInput)   
out2 = model2(singleInput)
#....
#outN = modelN(singleInput)

out = Concatenate()([out1,out2]) #[out1,out2,...,outN]
out = Dense(1, activation='sigmoid')(out)

model3 = Model(singleInput,out)

And if you already have all the models ready and don't want to change them, you can have something like this (not tested):

singleInput = Input((3,))
output = model3([singleInput,singleInput])
singleModel = Model(singleInput,output)

Define new input layer and use model outputs directly (works in functional api):

assert model1.input_shape == model2.input_shape # make sure they got same shape

inp = tf.keras.layers.Input(shape=model1.input_shape[1:])
model = tf.keras.models.Model(inputs=[inp], outputs=[model1(inp), model2(inp)])