Everything else is one-day spikes on regulatory reporting days, which of course can go in either direction, clear to the hard boundaries set by the penalty and deposit rates.

Yes, but the volatility is excessive. The ECB appears to be very reluctant to narrow the "corridor" between the deposit and marginal rates. They did so briefly (to ±50bp) at the height of the post-Lehman panic, and when they last lowered the repo rate they preferred to lower the deposit rate to zero rather than narrow the corridor to 0.25 - 0.75 - 1.25, for instance.

As to

does this look like the ECB has been targeting a rate?

Yes. You can tell that the ECBuBa is not targeting monetary aggregates

in his paper, Fullwiler writes:

Because the demand for reserve balances is very interest inelastic on a daily basis (where payment needs dominate the demand for reserve balances) or at least by the end of the maintenance period (where reserve requirements dominate), supplying more or fewer reserve balances than banks in the aggregate desire to hold will simply result in the interbank rate falling to the rate banks earn on balances in reserve accounts (if too many balances are supplied) or rising to the penalty rate assessed on overdrafts from the central bank (if too few are otherwise supplied). As such, a reserve balance "target" would be actually a de facto interest rate target at either the rate paid on balances in reserve accounts or the central bank's penalty rate. In practice, a reserve balance operating target would send the interbank rate fluctuating between these two rates, as banks' demand for reserve balances can shift significantly from day-to-day (depending upon the particulars of the national payments settlement system and the reserve requirement regime).

Significant volatility in the overnight rate is not desirable, however. As a member of the Fed's Board of Governors explained,

A significant increase in volatility in the federal funds rate would be of concern because it would affect other overnight rates, raising funding risks for most large banks, securities dealers, and other money market participants. Suppliers of funds to the overnight markets, including many small banks and thrifts, would face greater uncertainty about the returns they would earn and market participants would incur additional costs in managing their funding to limit their exposure to the heightened risk. (Meyer 2000, 4).

Even within neoclassical economic theory, such volatility in the overnight rate would become problematic from a monetary policy perspective "if [it were] transmitted to maturities which are deemed directly relevant for decisions of economic agents (Bindseil 2004, 100-101).

In ordinary times, the ECB appeared to be targeting "above but close to the repo rate", but it wasn't doing a very good job of it since it tolerated huge volatility on end-of-period days, all the way to the limits of the "corridor". In 2009, the ECB flooded the system with reserves when it switched from fixed-tender to fixed-rate-unlimited-tender weekly refinancing, and the interbank rate collapsed to slightly above the deposit rate, and the ECB didn't mind. So either the ECB changed its interest-rate target from 4% to 0.25% over the course of 2009 and decided to leave the refinancing rate at 1% just for laughs, or we have to conclude that the ECB is not, in fact, targeting a rate. The tachycardia period shows the extreme tolerance of the ECB for overnight interest rate volatility, again inconsistent with EONIA targeting.

If you are not convinced, try it on someone who has not been entirely debauched by economics. — Piero Sraffa