With 32Red group, the policy is one welcome bonus across all 4, and although you can have accounts at all 4, you cannot take another welcome bonus without their prior invite or permission (unless you fancy a WR of 100x).
Certain free tournaments offered by 32Red themselves are also restricted to one entry across the group, even if you have separate aliases for the 3 casinos offering it. This will be stated in the tournament rules on the email or website, but NOT the standardised rules as seen in the tournament lobby. This is to address complaints that some players use this trick to gain multiple entries, which is unfair on other players.
A similar welcome bonus policy exists in many casino groups, and seems to be becoming "industry standard". Once a loyal player though, you can take bonuses as they are offered provided this is within the rules. Groups like Casino Rewards for example will have a rule in their weekly bonuses of "this bonus is available in any one account across the group", even though it is offered in all the lobbies of all your casinos at the group.