No! You do NOT need demat account / trading account, to invest in mutual funds. In fact, you should strive to avoid Demat mode of holding when it comes to mutual funds.
Mutual funds are sold by AMC (Asset Management Companies) directly to you. RTA (Registrar and Transfer Agent) of the relevant AMC processes the transaction, and keeps records of your units purchase or selling.
It's already stored electronically, with legal ownership that it belongs to you.
For instance, if you buy units in SBI Bluechip (not a recommendation, using as an example), then SBI AMC would issue you units, and its RTA, CAMS, would be responsible for maintaining these transaction details as source of truth, when it comes to your holdings in the fund.
If you add Demat on top of that, it'd create an extra layer of abstraction, that adds plenty of headaches, with little to no benefits.
Demat folios (folios where your units are purchased through a broker in demat mode), don't let you port these into other platforms. In non-demat mode of holding (also known as Statement of Account mode, or SOA mode); your units are only with RTA, and you can purchase through one platform (INDMoney, for example), and sell those units through another platform (MFUtility, for example); while track your portfolio in a third platform (Kuvera, for example).
You aren't limited by the platform in this scenario. You can transact on your folio through any platform of your choice, track your holdings in any other platform of your choice. If a platform doesn't offer a functionality (say, instant redemption from Liquid funds, or creating new folio for tax optimizations), you can directly use the web portal of the respective AMC to place that particular transaction in your folio.
It gives you a freedom so that you're not vendor-locked-in with your broker.
Demat mode of holding also adds extra charges, in addition to usual charges of mutual funds like expense ratio, for each transactions. Depository fee, annual maintenance fees for your demat account, withdrawal stamp duty etc. are examples of charges you'd typically incur, should you use Demat account.
The question should be what does Demat mode of holding MF units bring to table, that makes it worth it? and not the other way around.
Presently, platforms like Coin, HDFC Securities etc. offer mainly Demat mode of holdings, which are best avoided if possible, especially in favor of free direct plan platforms, that you can get if you just have a savings account in a bank.