My attempt at Dijkstra's Algorithm in Python 3

Those comments are very difficult to read.

j = list(set(j)) #someone warned that the problematic test case gives the same edge many times, so this is to remove duplicates

Format this properly:

# The problematic test case gives the same edge many times,
# so this is to remove duplicates
j = list(set(j))

Then remove fluff:

# Remove duplicates. Helps some test cases.
j = list(set(j))

In this case just remove the first comment, because list(set(...)) is already known to mean deduplicate. And remove the second since it's implied by the fact you've written the code.

j = list(set(j))

Another pair is

a = [] # for input purposes
b = [] # for output purposes

Don't do this. Just call them

descriptive_name_1 = []
descriptive_name_2 = []

so you don't need the comment in the first place.

Then there's q = .... So I wrack my brain thinking about what word contains q. But it turns out you just meant to write num_test_cases. What does that have to do with the letter q?

Worse than that is when you outright lie. What would you expect the variable node to contain? A node, right? Nope - it contains the total number of nodes.

And you write

node = 0

so we expect the total number of nodes to be 0 at that point, right? Nope, you immediately change your mind and write

node, edge = list(map(int, input().strip().split()))

But OK, fine. At least we know what nodes contains, then: multiple nodes, so a list (or other collection) of lengths of nodes.

Nope, it actually means exactly the same thing as node.

A similar comment goes for edge, which should be num_edges.

Then there's adj, which you label "adjacency matrix". Why not just call it adjacency_matrix then? But it's not even a dense matrix like that would imply, it's actually a list of per-node adjacencies, so call it adjacencies.

Then there's starting and start, which are the same thing named differently. Call them start_node or at least be consistent.

Going back to b, you have

b = []

but again this is a lie, because actually

b = shortest(start_node, num_nodes, adjacencies)

So just write that.

distances = shortest(start_node, num_nodes, adjacencies)

For a, initialize it as close to usage as possible and call it edges.

If you write

for j in range(num_edges):

you're misleading me into thinking you're going to use the index. When you're not going to, write

for _ in range(num_edges):

And what's up with that, followed by

for j in edges:

followed by

for j in adjacencies:

Is j just your general "this is a thing" variable? Most people use x, elem and item for that. Instead, write

for _ in range(num_edges):
for edge in edges:
for adjacent in adjacencies:

The middle can be

for start, end in edges:

The comment

#edges are 6

Isn't coherent. Try

# Edges are length 6

Instead, though, don't put this information in shortest - use 1 and multiply out when using the results.

Don't call variables temp. It's a poor name.

You've double-indented the part where you update the neighbours.

Note that

num_nodes, num_edges = list(map(int, input().strip().split()))

is just

num_nodes, num_edges = map(int, input().strip().split())

Why do you do this?

for i in adjacencies[current-1]:
    neigh = i

Just write

for neighbour in adjacencies[current-1]:

And the content can be

for neighbour in adjacencies[current-1]:
    distance[neighbour-1] = min(distance[neighbour-1], distance[current-1] + 1)

Do you not think this whole -1 thing is getting a bit silly? When reading, just decrement the edge values and starting edge correctly:

edges = []
for _ in range(num_edges):
    start, end = map(int, input().strip().split())
    edges.append((start - 1, end - 1))

start_node = int(input()) - 1

while loops and if statements don't need their condition parenthesized, and operators should be spaced properly:

while (len(unique)>0):   # before
while len(unique) > 0:   # after

Then this is better as just while unique.

Don't put spaces before the parentheses in function calls or indexing.

minDist should be min_dist. You can also start with it set to num_nodes, and the same for distance's initialization.

unique is a horrible name, and has nothing to do with the variable's purpose. Try unvisited instead. It can be initialized as

unvisited = [i for i, adj in enumerate(adjacencies) if adj]

distance, unsurprisingly, does not mean distance but instead distances. Fix that.

This code:

min_dist = num_nodes
for i in unvisited:
    if distances[i] <= min_dist:
        min_dist = distances[i]
        distIndex = i
current = distIndex

is just

min_dist = num_nodes
for i in unvisited:
    if distances[i] <= min_dist:
        min_dist = distances[i]
        current = i

and current should be called closest. It is simplifiable to

closest = min(unvisited, key=lambda x: distances[x])

This is faster if you write

closest = min(unvisited, key=distances.__getitem__)

but the difference isn't vital.

One last thing with shortest - rename it to shortest_distances.

Put the rest of the code in a main function.

If you write

def read_pair(decrement):
    x, _, y = input().partition(" ")
    return int(x) - decrement, int(y) - decrement

then you can initialize num_nodes, num_edges and edges with

num_nodes, num_edges = read_pair(0)
edges = [read_pair(1) for i in range(num_edges)]

Note my use of partition instead of split as IMO it's a better description of the operation here.

Then

for adjacent in adjacencies:
    adjacent = list(set(adjacent))

doesn't actually work! adjacent = only affects the local scope! Instead, you want

adjacencies = [list(set(adj)) for adj in adjacencies]

which is even better as

adjacencies = [set() for _ in range(num_nodes)]
for start, end in edges:
    adjacencies[start].add(end)
    adjacencies[end].add(start)

as you never have the large intermediates then, and there's no real point converting back from a set.

This stuff:

distances.remove(0)
for i in range(len(distances)):
    if distances[i] == num_nodes:
        distances[i] = -1
    else:
        distances[i] *= 6
print(" ".join(str(e) for e in distances))

is just

print(*(-1 if i == num_nodes else i * 6 for i in distances if i))

Nice, that seems to have made it fast enough, but more important the code is more readable.

We can improve speed further by avoiding input, using sys.stdin directly to get buffering.

def main(lines):
    def read_pair(decrement):
        x, _, y = next(lines).partition(" ")
        return int(x) - decrement, int(y) - decrement

    ...

main(sys.stdin)

Note that this change is sufficient on its own to beat the time limit: you can apply it to your original code to pass the test. Note also that you shouldn't mix input with next; at least on Python 2 this throws an error saying

ValueError: Mixing iteration and read methods would lose data

Though this error is gone in Python 3, that doesn't leave me feeling good about it.

I'd finish off with

from __future__ import print_function

to make it Python 2 compatible.

from __future__ import print_function

import sys

def shortest_distances(start_node, num_nodes, adjacencies):
    distances = [num_nodes] * num_nodes
    unvisited = [i for i, adj in enumerate(adjacencies) if adj]

    distances[start_node] = 0
    while unvisited:
        closest = min(unvisited, key=distances.__getitem__)
        unvisited.remove(closest)

        # Update the distances of each neighbour
        for neighbour in adjacencies[closest]:
            distances[neighbour] = min(distances[neighbour], distances[closest] + 1)

    return distances

def main(lines):
    def read_pair(decrement):
        x, _, y = next(lines).partition(" ")
        return int(x) - decrement, int(y) - decrement

    num_test_cases = int(next(lines))
    for i in range(num_test_cases):
        num_nodes, num_edges = read_pair(0)
        edges = [read_pair(1) for i in range(num_edges)]
        start_node = int(next(lines)) - 1

        adjacencies = [set() for _ in range(num_nodes)]
        for start, end in edges:
            adjacencies[start].add(end)
            adjacencies[end].add(start)

        distances = shortest_distances(start_node, num_nodes, adjacencies)

        print(*(-1 if i == num_nodes else i * 6 for i in distances if i))

main(sys.stdin)