No. Not even with Cat6a.
In terms of characteristic impedance, things don't look too bad. Both the single-ended and differential impedance specifications for Cat5e sit within the tolerance range specified for USB 3.0 Gen1. However, when you look into the signal specifications, it quickly becomes apparent that Cat5e is wholly unsuitable.
1000BASE-T achieves bidirectional gigabit speeds through the use of a PAM5 line code across 4 pairs, with some echo cancellation wizardry to achieve bidirectional communications. The PAM5 line code has five distinct signal levels that can be transmitted on the line, which allows for 2 bits to be encoded per symbol. This means you have four channels at 125MBd with two bits per symbol, i.e. 1000Mbps total. I'm handwaving over heaps of details here (Trellis coded modulation, adaptive equalisation, etc.) but for this purpose they're not super relevant. What is relevant is that the bandwidth of each channel is 62.5MHz, and Cat5e cabling is rated to 100MHz.
USB specs and speeds are a whole thing, with extremely confusing naming conventions and changes throughout the standard's history. I'm going to assume that we're talking about USB 3.0 SuperSpeed 5Gbps.
USB 3.0 SuperSpeed does not use fancy multi-level signalling for data transport, nor does it transmit and receive over the same lines simultaneously. It instead uses NRZI. Non-return-to-zero has two signal states - positive and negative - which are characterised by one line in the pair having a positive voltage and the other line in the pair having a corresponding negative voltage. The inverted variant of NRZ, i.e. NRZI, uses the absence or presence of a signal state transition at each clock cycle to signal a 1 or 0 respectively. USB 3.0 SuperSpeed has two channels (two differential pairs) with one used for transmit and the other used for receive. This is full-duplex, so the full 5Gbps data rate is used on both lines. The minimum bandwidth for NRZI is half the signal rate, so each of the USB 3.0 SuperSpeed differential pairs operates with a 2.5GHz bandwidth.
This means that the bandwidth required per pair on USB 3.0 SuperSpeed is 40 times as high as the bandwidth required per pair in 1000BASE-T networking.
Even if the cabling was suitably rated for the signal transport, your connector would also need to be assembled in a way that does not introduce significant impedance discontinuities, and the lines would need to be matched to keep within the inter-pair and intra-pair timing skew budgets.
Cat6a is characterised up to 500MHz, which is still 5 times too little for USB 3.0 SuperSpeed. You could theoretically run USB 2.0 High Speed connection over Cat6a, since it only operates at 480Mbps.
Power delivery would also be an issue. Each twisted pair in Cat5e cable is usually rated for 960mA, in order to support 802.bt Type 4 PoE, but the maximum cable resistance specifications are set based on the expectation of a PoE voltage in the range of 37-57V, not 5V. You might have cables where the total DC resistance for a pair is as high as 12.5Ω, which at 500mA is a 6V drop. That's not a problem if your source voltage is at least 37V, like it is in PoE, but for a 5V source it's completely unworkable. You could probably get away with the USB 2.0 standard power limit of 100mA, using two pairs to carry VUSB and one pair plus the shield for carrying GND, with the remaining pair for D+/D-, giving you a max droop of 625mV (although probably less for shorter cables).
In summary: it's just not going to work.
