There is nothing wrong with keeping emergency fund and back fund together. But if you keep them together then you may face some problems like if you keep your emergency fund and back fund in the same account then if your account gets locked then in this situation if you face any emergency then you may face some problems.
My view on keeping your emergency fund and what you term as back up fund( which to me looks like the same thing) is that, it all depends on the individual involved and wether or not, keeping all the funds that's serving as a shield to your investment at the same spot is the best thing to do.
The main thing is to know that you're never spending your emergency fund or back up fund for no just reason. Once you've set that boundary for yourself, regardless of wether you keep the funds with another fund or you keep it separately, you're certain that everything is intact and easily accessible at whatever time you will have need of it.
Emergency fund is money kept for personal or family needs and back fund is money kept for future in any personal or business sector. If you do not have proper financial management skills then you may spend all your money in emergency fund or back fund. So it is better to keep emergency fund and back fund separately.
If you do your budgeting rightly, you will have little to no issues regarding how you utilise your funds. I wouldn't be too comfortable referring funds that's meant for my upkeep as emergency fund, the reason is that it's expected that while doing budgeting, the first item should be proper allocation for;
1. Upkeep
2. Investment budget,
3. and then emergency fund
4 and additionally, there might be further provision for funds you can use in buying the DIP depending on the stage you are as an investor.
Funds for family expenses should be given adequate priority even above your emergency fund and your investment fund because they are what help you remain invested for the long run.