Your problem is a graph-walking problem of finding connected subgraphs. It is a little more challenging because your data structure has two types of nodes ("companies" and "pubishers") rather than one type.
You can solve this with a single recursive CTE. The logic is as follows.
First, convert the problem into a graph with only one type of node. I do this by making the nodes companies and the edges linkes between companies, using the publisher information. This is just a join:
select t1.company as node1, t2.company as node2
from table1 t1 join
table1 t2
on t1.publisher = t2.publisher
)
(For efficiency sake, you could also add t1.company <> t2.company but that is not strictly necessary.)
Now, this is a "simple" graph walking problem, where a recursive CTE is used to create all connections between two nodes. The recursive CTE walks through the graph using join. Along the way, it keeps a list of all nodes visited. In SQL Server, this needs to be stored in a string.
The code needs to ensure that it doesn't visit a node twice for a given path, because this can result in infinite recursion (and an error). If the above is called edges, the CTE that generates all pairs of connected nodes looks like:
cte as (
select e.node1, e.node2, cast('|'+e.node1+'|'+e.node2+'|' as varchar(max)) as nodes,
1 as level
from edges e
union all
select c.node1, e.node2, c.nodes+e.node2+'|', 1+c.level
from cte c join
edges e
on c.node2 = e.node1 and
c.nodes not like '|%'+e.node2+'%|'
)
Now, with this list of connected nodes, assign each node the minimum of all the nodes it is connected to, including itself. This serves as an identifier of connected subgraphs. That is, all companies connected to each other via the publishers will have the same minimum.
The final two steps are to enumerate this minimum (as the GroupId) and to join the GroupId back to the original data.
The full (and I might add tested) query looks like:
with edges as (
select t1.company as node1, t2.company as node2
from table1 t1 join
table1 t2
on t1.publisher = t2.publisher
),
cte as (
select e.node1, e.node2,
cast('|'+e.node1+'|'+e.node2+'|' as varchar(max)) as nodes,
1 as level
from edges e
union all
select c.node1, e.node2,
c.nodes+e.node2+'|',
1+c.level
from cte c join
edges e
on c.node2 = e.node1 and
c.nodes not like '|%'+e.node2+'%|'
),
nodes as (
select node1,
(case when min(node2) < node1 then min(node2) else node1 end
) as grp
from cte
group by node1
)
select t.company, t.publisher, grp.GroupId
from table1 t join
(select n.node1, dense_rank() over (order by grp) as GroupId
from nodes n
) grp
on t.company = grp.node1;
Note that this works on finding any connected subgraphs. It does not assume that any particular number of levels.
EDIT:
The question of performance for this is vexing. At a minimum, the above query will run better with an index on Publisher. Better yet is to take @MikaelEriksson's suggestion, and put the edges in a separate table.
Another question is whether you look for equivalency classes among the Companies or the Publishers. I took the approach of using Companies, because I think that has better "explanability" (my inclination to respond was based on numerous comments that this could not be done with CTEs).
I am guessing that you could get reasonable performance from this, although that requires more knowledge of your data and system than provided in the OP. It is quite likely, though, that the best performance will come from a multiple query approach.
