For analysis, it's essentially a problem of 1) non-independence of observations, and 2) weighting of the multiple control groups in the analysis. Below is assuming all of your matched cases from both control groups (combined) are unique observations--not duplicated.
The simplest solution, which is probably reasonable acceptable to most people, is likely to simply analyze the two sets of matched pairs separately, and present it as a "replication". If the results are the same for both matched groups, then you can say the same result is achieved with propensity score matching even with an independent control group. If the results are different, it would merit investigation as to whether the difference is due to detectable differences in the matched characteristics of the two control groups (or their difference-in-difference with the study group). Such differences could emerge, and be observed, for example, if a first iteration of the propensity match is a "closer match" than the second iteration of propensity match.
Partial solutions, which I'm sure would be unsatisfactory to the scientific community (and to me), would be to: a) code the two control groups (e.g. 1 vs. 0) into a variable and enter it as a covariate in a model using all the data, including the double study observations. This would not solve the problem of non-independence, but would more-or-less ensure both control groups are weighted equally and fairly, removing any bias that snuck in from one propensity match iteration to the next. b) Pool or average your responses from the two control groups into a single observation, as to be matched with a single observation of the study group. This resolves the issue of non-independence, and partially ensures the control observations are weighted relatively equally; however, it definitely distorts the underlying true variance of observations in the control groups (which wouldn't necessarily be a problem if you had, say, 20 matched control groups, and could use their observed variance).
A slightly more complicated solution would be to construct a multilevel model. There, you can enter the study group ID as the identifier for level-2, and the control group ID's as independent observations for level-1. Since you would presumably be predicting level-2 outcomes from level-1 predictors, you could use a simple fixed-effects model (which is relatively simple, nice). This should take care of both your issues of non-independence and comparative weighting of the multiple control groups in a single analysis. Also, as in a) above, you could again code and enter propensity match iteration as a covariate in the model.
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