Experimental design with two groups, pre-post measurement, and where everyone is treated
I would call it a pre-post quasi-experiment, albeit where you are assessing the intervention in two different pre-existing groups.

In terms of assessing the effect of the intervention, there are more threats to causal inference, when you don't have a control group (i.e., one where you have pre and post measures but with no intervention or a control intervention). I.e., In addition to the effect of the treatment, there are many other explanations for any changes observed pre-post. For example, learning, maturation, fatigue, and so on. Background knowledge may be able to assist you in appraising which if any of these are likely to be significant.

You then have two observed groups. This aspect is more like an observational study. You are assessing the interaction between group and the treatment. Because group is an observational variable, you would want to take care in attributing causal explanations to any differences between groups in treatment effect to group membership. Nonetheless, often in this context, the interest is more about understanding how well a treatment generalises to different populations.

So in summary, It would be better if there was a control group. Ideally, you would have a fully crossed design. I.e., group (A and B) by control (treatment and control) where participants were randomly allocated to treatment or control. But if that's not possible, the data would still be interesting; you'd need to think carefully about carry-over effects.