Did you ever stop to think that there's no possessive form of That? There is, on the other hand, a possessive case of who.

That's because the two words have been intertwined since the days of Middle English, and sometimes merge imperceptibly together:

WHAN that Aprille with his shoures soote
The droghte of Marche hath perced to the roote,
And bathed every veyne in swich licour,
Of which vertu engendred is the flour;
Whan Zephirus eek with his swete breeth     5
Inspired hath in every holt and heeth
The tendre croppes, and the yonge sonne
Hath in the Ram his halfe cours y-ronne,
And smale fowles maken melodye,
That slepen al the night with open ye,         10
(So priketh hem nature in hir corages:
Than longen folk to goon on pilgrimages,
And palmers for to seken straunge strondes,
To ferne halwes, couthe in sondry londes;
And specially, from every shires ende           15
Of Engelond, to Caunterbury they wende,
The holy blisful martir for to seke,
That hem hath holpen, whan that they were seke.
Bifel that, in that sesoun on a day,
In Southwerk at the Tabard as I lay             20
Redy to wenden on my pilgrimage
To Caunterbury with ful devout corage,

‘That’ is a very complicated word, because it can be used to refer to both people and things, as a way to indicate proximity or particularity when contrasted with ‘this,’ or even attitude.

Compare and contrast:

Who's this woman?

Who's that woman?

By distinguishing between this and that, one may imply an emotional distance as well as mere proximity, but there's a somewhat similar distinction between that and who, so there's a sort of ‘usage rule,’ actually more of a guideline, that when referring to people, one uses ‘who,’ and when referring to things, one uses ‘that.’

I saw a man who was dressed all in green the other day, so I figured that he must be a leprechaun.

I saw a fire hydrant that was painted all in green the other day, so I figured that it must be a leprechaun in disguise.

Notice that ‘that’ and ‘it’ have a tendency to pair. You can usually decide which one to use based upon whether one is at all tempted to refer to any particular entity as ‘it.’

Because there is no possessive case for ‘that,’ ‘who’ takes over when it's important to indicate possession. So one might refer to:

The soulless corporation that tosses widows and orphans out on the street if they fall even slightly behind on on paying rent,

but to:

The soulless corporation whose policies value profits over human life.

Despite the fact that one might want to indicate an emotional distance, one is forced to personalise when one talks casually of ownership, although of course we can avoid the little word through more elaborate circumlocutions:

The soulless corporation which has policies that value profits over human life.

The souless corporation of which there are many complaints of inhumanity and cruelty, is run by Bozo, the Clown.

English is a very flexible language, but please note that ‘of which’ can easily be heard as stilted and/or awkward, although it's theoretically available as a sort of possessive that avoids the use of ‘whose,’ and in fact has been available since Chaucer's day, as plainly seen above....

It's usually heard (and written) nowadays in more-or-less trite phrases like:

He has his faults, of which there are many.

None-the-less, despite this flexibility, there are many people who will be discomfited when they read sentences like this:

My sister, that loves cats, has managed to crowd seven of them into her tiny apartment.

In a sentence like that, the word ‘that’ in relation to a human being would be objectifying in a manner that would be ever-so-slightly jarring to many of us.