Can I day trade with €5,000 of savings?

If that $5k is your entire savings, you would do well to work on budgeting your income vs expenses, and advance your career through whatever means necessary. That will likely have a far greater impact on your financial position in the long term.

Day trading is incredibly risky. You say you are doing this partially 'as a form of entertainment'. Can you think of a hobby that would cost you less than $5000? Be very, very careful that you don't get in over your head.

I'm a bit confused about your question because specified Euros along with a tag of Europe. Is there a Pattern Day Trader Rule there or are asking about trading in the US? Be that as it may, here's the scoop for the US.

You can make as many round trip day trades as you want with settled cash in a Cash Account. With $5k in the account, you could make one $5k round trip day trade or five round trip $1k trades. You would then have to wait two days (T+2) for the cash to settle before you would be able to trade again.

The PDT rule would be an issue if you were doing this in a margin account. At 50% Reg T margin you could trade $10k but you would have the 3 trade limit. The rule limits you to three or more day trades (options and equities) in a rolling FIVE business day period in a MARGIN ACCOUNT, provided the number of day trades are more than six percent of the customer's total trading activity for that same five-day period.

You could split your money into two accounts. Some issues would be less money available for trading in each account, more tax paper work, and the inconvenience of toggling between accounts and keeping track of the number of trades if using margin.

Commissions would not increase unless you were splitting your money into smaller parcels in order to make more trades. With most major discount brokers having eliminated commissions in the US, this isn't an issue.

If you make more than the aforementioned 3 trades then you will be labeled a Pattern Day Trader and you must meet the $25k minimum account equity at all times. If you violate this, you will not be permitted to day trade until the account is restored to the $25k minimum equity level. You will have at most, 5 business days to deposit funds to meet the call. Until the call is met, day-trading will be restricted to the standard two times (50%) margin requirement (A PDT is allowed to trade at 25% margin intraday unless the brokers has a higher requirement. If the margin call is not met by the fifth business day, the account will be further restricted to trading only on a cash available basis for 90 days or until the call is met.

Day trading can be very profitable but anecdotally, something like 90% of wannabe get rich quick traders lose money. I'd suggest that you paper trade for the entertainment and use the €5,000 for something tangible that you can enjoy (Vacation? Down payment on a car? Etc.). Noobs tend to donate their money quickly :->)

In addition to Bob Baerker's excellent answer:

It largely depends on how much you're paying in trading fees. Assuming you're paying $5 in fees per trade and you're trading 10% of your total investment ($500) you're losing 1% for every single trade. So a buy and a sale of the same stock leaves you down 2% if the stock doesn't move. If you're thinking about putting your whole deposit into a single trade it gets slightly easier to actually make a profit but also makes it far more likely that you'll go broke very quickly.

If you don't have trading fees and you don't have legal restrictions about how many trades you can execute in a day you can actually start trading with $5. Which is also something I would strongly recommend: Start with money you can easily write off and see how that goes.