How subscriber table updated during snapshot replication?
Is it really how it is supposed to work ?

Yes this how it is supposed to work. You only run snapshot at the beginning of  setting up transactional replication. Unless you are out of sync due to any error you do not need to run snapshot again. If you are running snapshot replication ONLY you can control when you want to resync all your articles and nothing should connect to subscriber while snapshot is being applied.

From books online:


  Snapshot replication is most appropriate when data changes are
  substantial but infrequent. For example, if a sales organization
  maintains a product price list and the prices are all updated at the
  same time once or twice each year, replicating the entire snapshot of
  data after it has changed is recommended. Given certain types of data,
  more frequent snapshots may also be appropriate. For example, if a
  relatively small table is updated at the Publisher during the day, but
  some latency is acceptable, changes can be delivered nightly as a
  snapshot.
  
  Snapshot replication has a lower continuous overhead on the Publisher
  than transactional replication, because incremental changes are not
  tracked. However, if the dataset set being replicated is very large,
  it will require substantial resources to generate and apply the
  snapshot. Consider the size of the entire data set and the frequency
  of changes to the data when evaluating whether to utilize snapshot
  replication.


When is a subscription available; when can the subscription database be used?


  A subscription is available after the snapshot has been applied to the
  subscription database. Even though the subscription database is
  accessible prior to this, the database should not be used until after
  the snapshot has been applied. Use Replication Monitor to check the
  status of snapshot generation and application.


Are there settings somewhere to avoid such behaviour ?

If you are sure that your schema and data is already in sync or you only want future changes to get replicated you can use following option to avoid snapshot. I used this many time during upgrade with downtime.

@sync_type=replication support only


  Provides automatic generation at the Subscriber of article custom
  stored procedures and triggers that support updating subscriptions, if
  appropriate. Assumes that the Subscriber already has the schema and
  initial data for published tables. When configuring a peer-to-peer
  transactional replication topology, ensure that the data at all nodes
  in the topology is identical. For more information, see Peer-to-Peer
  Transactional Replication.
  
  Not supported for subscriptions to non-SQL Server publications.