I always thought that having a KYC verified account would make the process of recovering such funds easier. What about a death certificate proving that he died issued by an official government institution and a document proving that his wife and/or children are the legal inheritors of his assets?
What happens with the money? The exchange just takes possession of it eventually?
This depends on an exchange. Exchanges that follow laws to the letter like coinbase make easier for one to recover the funds unlike less regulated exchanges like Binance.
Lets take a look at the terms and conditions of the 2 exchanges OP talks about;
1. Bittrex
Talks nothing about death of account holder, they talk only about unclaimed property -
https://bittrex.zendesk.com/hc/en-us/articles/3600005608718.8. Unclaimed Property
In certain circumstances, Bittrex may have an obligation to report any Tokens in your Hosted Wallet or any fiat currency in your Bittrex Account to the applicable governmental entity as unclaimed property, such as where your Bittrex Account has been inactive for a period of several years (as defined by the relevant state statutes) or where Bittrex ceases providing Services. If this happens, Bittrex will attempt to contact you using the contact information provided by you. If you do not respond, Bittrex may be obligated to turn over any Tokens in your Hosted Wallet or any fiat currency in your Bittrex Account to the applicable governmental entity after deducting any fees payable to Bittrex. The applicable governmental entity may require Bittrex to liquidate any Tokens in your Bittrex Account into fiat currency and turn over the resulting fiat currency to the governmental entity. You agree that Bittrex will not have any responsibility or liability for any losses, damages, or other harm that you may incur in connection with Bittrex turning over Tokens or fiat currency to the applicable governmental entity or Bittrex’s liquidation of the Tokens.
2. Coinbase
Talks about death of account holder and how one can claim the assets -
https://www.coinbase.com/legal/user_agreement/payments_europe13.11 Death of Account Holder. For security reasons, if we receive legal documentation confirming your death or other information leading us to believe you have died, we will freeze your Coinbase Account and during this time, no transactions may be completed until: (i) your designated executor / trustee has opened a new Coinbase Account, as further described below, and the entirety of your Coinbase Account has been transferred to such new account, or (ii) we have received proof in a form satisfactory to us that you have not died. If we have reason to believe you may have died but we do not have proof of your death in a form satisfactory to us, you authorize us to make inquiries, whether directly or through third parties, that we consider necessary to ascertain whether you have died. Upon receipt by us of proof satisfactory to us that you have died, the executor / trustee you have designated in a valid Will or similar testamentary document will be required to open a new Coinbase Account. If you have not designated an executor / trustee, then we reserve the right to (i) treat as your executor / trustee any person entitled to inherit your Coinbase Account, as determined by us upon receipt and review of the documentation we, in our sole and absolute discretion, deem necessary or appropriate, including (but not limited to) a Will or similar document, or (ii) require an order designating an executor / trustee from a court having competent jurisdiction over your estate. In the event we determine, in our sole and absolute discretion, that there is uncertainty regarding the validity of the executor / trustee designation, we reserve the right to require an order resolving such issue from a court of competent jurisdiction before taking any action relating to your Coinbase Account. Pursuant to the above, the opening of a new Coinbase Account by a designated executor / trustee is mandatory following the death of a Coinbase Account owner, and you hereby agree that your executor / trustee will be required to open a new Coinbase Account and provide the information required under Section 3 of this Agreement in order to gain access to the contents of your Coinbase Account.
Binance despite being the biggest exchange, doesn't day anything about Inactive accounts, unclaimed property or death of account holder
is this your account or your uncle account 
