Past tense and possession are two different jobs in English
English marks past tense with its own machinery. The regular pattern is the -ed ending, which serves both the past tense and the past participle, as in a comparative description of English and Hausa morphology [2]. Irregular verbs do their own thing, and the auxiliary verb to do has past tense forms that standard English treats separately from the main verb to do [3]. None of this involves possession.
Possession is marked differently. English uses possessive pronouns such as my, your and our, and in writing the possessive case is signalled with the apostrophe [1][4]. A possessive pronoun tells you who owns something; a past-tense ending tells you when something happened. They are not the same kind of signal, and one does not do the other's work.
Why the two can look tangled
The confusion is easy to understand once you look at spelling. The apostrophe appears in the possessive case, but it has also turned up in the regular past tense in historical writing — one study of the apostrophe's history counted thousands of past-tense forms written with an apostrophe alongside far more written without one, and thousands of possessive instances as well [4]. That is a story about how a punctuation mark was standardized, not about grammar merging two categories.
Regional grammar adds another layer of apparent overlap. In southern British English, past tense forms are sometimes extended to the past participle, and possessive pronouns appear in the same discussion of endings [1]. In Tyneside and Northumbrian English, reflexives are formed by adding self or selves to possessive forms, with the past tense form was noted as an exception to a pattern [5]. These are local quirks of particular varieties, and they show possessive material and past-tense material appearing near each other — not one being used to build the other.
So is English unusual here?
The premise of the question doesn't quite hold: there is no general rule in English that possession marks past tense. What the evidence describes is a language where past tense is carried by verb endings and irregular forms [2][3], possession is carried by pronouns and the possessive case [1][4], and the two occasionally brush against each other in spelling history or in the grammar of specific regional varieties [1][5].
That makes English unremarkable on this point rather than strange. The interesting variation the papers document is local: how the apostrophe was standardized across possessive and past-tense uses [4], and how particular dialects form reflexives or extend past-tense forms [1][5]. If you are looking for a genuine oddity, the regional patterns are where to look — not in the core tense system.
Sources used in this answer
The grammar of southern British English
Southern British English shows past tense forms extended to the past participle, and possessive pronouns feature in the discussion of endings.
Inflectional Process And Errors In English And Hausa Languages: A Comparative Analysis
English uses the -ed inflection for both past tense and past participle and has a distinct possessive case, as compared with Hausa.
International English: A guide to the varieties of standard English
Standard English does not distinguish past tense forms of auxiliary do from those of main verb do, and possessive pronouns are discussed separately.
'The night before beg'd ye queens's pardon and his brother's': the apostrophe in the history of English
Historical counts show the apostrophe appearing in regular past tense forms as well as in the possessive case during its standardization.
The grammar of Tyneside and northumbrian English
In Tyneside and Northumbrian English, reflexives are formed from possessive forms, with the past tense form was noted as an exception.