No. For equivalent quality, even a high-order delta-sigma encoder will need more bits than PCM.

The ratio between the sampling frequencies is greater than the ratio between the number of bits per sample. How much greater it is depends on the order of the sigma-delta modulator and higher-order modulators need much less extra bits, but they have other problems, which prevent the increase of the order too much.

Unlike PCM, delta-sigma modulation does not provide any guarantee about the maximum error at a given point in time, i.e. about the maximum instantaneous difference between the original signal and the encoded signal. Delta-sigma modulation only guarantees that certain statistical properties of the original signal and of the encoded signal are the same. For example in the case of the first-order delta-sigma modulator it is guaranteed only that the difference between the integrals over an interval of time of the original and of the encoded signals is smaller than a superior limit that depends on the length of the time interval.

This is why even for high-order delta-sigma modulators the ratio between their sampling frequencies and the sampling frequency of an equivalent PCM signal must remain great, because the sampling period of the PCM signal must be long enough so that the statistics of the delta-sigma signal accumulated during it make sense.

I do not like the term "noise shaping" and it is somewhat misleading, because it is not an intrinsic property of the delta-sigma modulator, but in order for the noise to be "shaped", it must actually exist. That means that when the input signal does not include any noise, the output of the delta-sigma modulator will oscillate periodically and deterministically around a value and there will be no "noise shaping". However, real electronic circuits always have noise, and when that noise is insufficient, in delta-sigma ADCs additional noise is injected at input, to prevent the periodic oscillations at output.

A simple oversampler can be regarded in some sense as a zeroth order delta-sigma modulator, because some of the formulae for delta-sigma modulators apply to it when the order is set to 0.

Just oversampling cannot produce any improvement in the resolution of a PCM signal obtained by filtering the oversampled signal. But if white noise with an amplitude equal to the quantization interval is added at the input of the oversampler, then by averaging the output, i.e. by low-pass filtering it, additional bits per sample can be obtained depending on how long is the averaging interval. This is the well known statistics property that the average of N random samples will have a lower variance than the original samples, which decreases for increasing N.

The same happens with delta-sigma modulators, the difference is only that the number of additional bits of resolution per PCM sample grows much faster with the length of the averaging interval than in the case of the simple oversampler, the higher the order of the delta-sigma modulator, the more additional bits at a given averaging interval.

Nonetheless, while the simple formula that expresses the achievable signal-to-noise ratio, i.e. the number of bits per PCM sample, as a function of the averaging interval, looks like one could get infinite resolution by just increasing the order of the delta-sigma modulator, that does not work in reality, where the order is limited to rather low values and the formula for the additional bits of resolution is not valid for very short averaging times, when the higher-order statistics are meaningless.