No, it would not be correct to apply multigroup CFA because there is a dependency in the data due to having the same people take both test versions. But you can apply measurement models of longitudinal CFA and test measurement equivalence/invariance in analogy to how you would test it using multigroup CFA.

When the data for both languages come from the same group of participants, a multigroup confirmatory factor analysis (CFA) is appropriate. The aim of a multigroup CFA is to determine whether factor structures are the same across two or more groups. In your case, there are two test versions: English and Korean.

You can determine whether the factor structure of the test is equivalent across languages by testing for measurement invariance. As a result, you can evaluate the comparability of test scores across the two languages and make inferences about the test's validity.

When interpreting a multigroup CFA's results, it is important to consider other factors about the test and the population. Furthermore, it is essential to carefully examine the assumptions of the model and ensure that the data meets the requirements for conducting a CFA, including adequate sample size and reasonable levels of factor correlation.

Hassan Babapour Multigroup analysis is for independent groups. As you wrote, in this case, there is only one group ("the data for both languages come from the same group of participants"). Therefore, multigroup analysis is inappropriate because it does not address the dependencies in the data due to repeated assessments of the same people on multiple occasions (time points). Those dependencies can be properly addressed with CFA models for longitudinal data, for example, latent state models. See, for example, Figure 8 in the chapter below on longitudinal SEM:

Chapter Longitudinal structural equation modeling of personality data

Latent state models allow you to examine measurement equivalence in the same way as would be done in multigroup models for independent samples/groups.

Christian Geiser To determine the best course of action, it may be necessary to consider additional information and carefully assess the assumptions and requirements of the research design. This could include considerations such as the sample size, the level of factor correlation, and any other factors that may impact the validity of the results.