Before researching into harmonic distortion data and other fine matters as possible causes of your observation, make sure these do not fall victim of measurement artefacts. Doing measurements, when there are other sources of significant current pulses synchronous with your function generator signals, isolate your voltage follower circuit from any other circuits of which it may be a constituent part. If pushed to the brink of despair in vain search for truth, I would not hesitate to make a one-off PCB with only this circuit. Why? Your inductor may be a receiver of EMI. Then, the interference can leak through supply voltage: at 600 KHz the supply voltage rejection ratio decreases down to mere 60 dB. Also, it may be helpful to read [TLE2141 and TLE2141-Q1 EMI Immunity Performance][1]. 

Interference and supply leakage having been excluded as a result of consideration into the matter or the DUT arrangement reshaping following my recommendation, take care of the phase difference issue and the timing data: if the standard deviations of 15 for a phase shift and 14.4 MEG for the frequency shown on your scope display have a legitimate explanation, attach these to the question, if you are still interested in the quality answers from the community. If there is no legitimate explanation, take care of this matter! It may give a clue capable to solve your problem.

Do not discard the simulation data from consideration: however sound, speaking in general, is disbelieve concerning simulation predictions, I've examined the TI simulation model for TLE2141 and see no implementation details (at least those I am aware of) that the developers had missed to include. For the record: in my opinion, the model covering of harmonic distortion is quite trustworthy.

[![TLE2144-q639414][2]][2]

No visible distortion at all. You may become interested in output current:

[![TLE2144-q639414-current][3]][3]

The simulation run for a \$A_{VD}\$-vs-frequency graph:

[![TLE2144-q639414-ac(fig-15)][4]][4]

gives a picture which does not agree at low frequencies with Figure 15 of the datasheet. I do not know the feedback parameters of their test measurement circuit and failed to adjust feedback resistors for DC gain of 110 dB, the datasheet specifies only capacitive load parameter. On the other hand, the DC gain of 93 dB does agree with \$A_{VD}\$=220V/mV parameter shown in table **TLE2141 ELECTRICAL CHARACTERISTICS** of the datasheet, page 5. Anyway, \$A_{VD}\$(freq) is mostly used for stability analysis, and the simulation is in good agreement with the datasheet in this part.

  [1]: https://www.ti.com/lit/an/sboz026b/sboz026b.pdf?ts=1666594136154
  [2]: https://i.sstatic.net/euzDM.png
  [3]: https://i.sstatic.net/c1335.png
  [4]: https://i.sstatic.net/lIbAa.png