This is actually a Tory policy, where the law was passed by the previous Tory government and came into force under labour.

That's not a get out of jail card, because they could have stopped it, and instead choose to continue it, just like they continued the NHS anti trans policies. Or the way they choose to continue the same austerity budget rules.

At this point, I'd say they're exactly as right wing as the Tories.

Labour have always had a bad track record of civil liberties. Although as the other reply notes this was a Tory policy initially, I've not gone over the proposed amendments to see if they'd have implemented it slightly differently (I suspect not). Unfortunately policies like these are both popular and free for the govt to implement so it's literally a no-brainer for MPs to vote through without thinking about consequences.