While the combination of HRHHID and HRHHID2 uniquely identifies households within each sample month, they do not link households across their appearances in the CPS panel. For linking households across months, IPUMS provides the linking variable CPSID (CPSIDP and CPSIDV for linking individuals). It is important to note, however, that the CPS samples dwellings, and then interviews members of households within the sampled dwellings. If a family or household moves dwellings, they are not followed by the CPS; the new inhabitants of the sampled dwelling will take their place. The dwelling retains the new household-level linking key, CPSID, linking the new inhabitants to those who resided in the dwelling previously even if they are different individuals. This means that simply looking for changes in tenure at the person or household level over time will not only capture people who purchase the dwellings they live in and formerly rented, but people who purchased and moved into dwellings that another household was previously renting.
To identify individuals in the CPS who transitioned from renting their dwelling to owning that same dwelling during their CPS observation period, you should use CPSIDV in combination with OWNERSHP. CPSIDV is a unique person identifier that links individual respondents over their CPS panel. It is a validated identifier, meaning it only makes links between those records whose SEX and RACE values do not change and whose AGE values change in expected ways over time. Using CPSIDV, you can be sure that you are looking at the same individuals over time. Materials from our 2021 CPS Virtual Summer Data Workshop on linking the CPS may be of use to you. See, for example, this presentation on CPS linking and validation.
OWNERSHP reports whether the household rented or owned its housing unit. Regardless of who within the household owned or rented the housing unit, all members of the household will share the same value of OWNERSHP. Therefore, it is okay to study OWNERSHP at the person level using CPSIDV to link individuals, rather than at the household level.
You cannot capture individuals who transitioned from renting their dwelling to owning and living in another dwelling. I expect the number of individuals who make the transition you are interested in (the former) to be very small. With very small sample sizes, your standard errors will be large, and it may be impossible to infer meaningful information. See this IPUMS Forum thread for a quick discussion of small sample sizes.