I don't think there are. There are several other ground based tracking and satellite relay systems, for example the Deep Space Network (DSN), Indian Deep Space Network (IDSN), Chinese Deep Space Network, Soviet Deep Space Network and of course the European Space Tracking (ESTRACK), but these are all ground based, and none of them exclusively commercial in nature. I wasn't able to find any mention of a commercial, or even a non-commercial and already existing space-to-space tracking and relay system similar to NASA's Tracking and Data Relay Satellite System (TDRSS)
NASA Goddard Space Flight Center (GSFC) however has another study active, called Space Based Relay Study (SBRS), under the umbrella of the Goddard's Exploration and Space Communications Division on behalf of the Space Communications and Navigation Program (SCaN), tasked with identifying future space-based relay communication and navigation architectures to support NASA missions in the 2022 and beyond timeframe. Since the NASA pages are down due to US government shutdown, I wasn't able to find any information if this is meant to replace TDRSS, or co-exist alongside it, or if it is meant to also provide commercial services. But I would presume not, and it will be a gradual replacement of NASA's own Tracking and Data Relay Satellites (TDRS).
Additionally, European Space Agency (ESA) has its own plans to build a European Data Relay System (EDRS) with a constellation of GEO (Geosynchronous Orbit) satellites intended to relay information and data between satellites, spacecraft, Unmanned Aerial Vehicles (UAV) and ground stations, currently under implementation within the Advanced Research in Telecommunications Systems Programme (ARTES-7):
After a successful demonstration of the Semiconductor-laser
Inter-satellite Link EXperiment (SILEX) in 2001 between Artemis GEO
satellite and SPOT-4 LEO satellite, EDRS will become the first
commercial European data relay system providing a wide range of
operational services (both optical and Ka-band based).
Source: Status of the European Data Relay Satellite System, ICSOS, October 9, 2012 (PDF)
First customer of the EDRS service (using Sentinel 1A/B and 2A/B LEO satellites) will be Copernicus - The European Earth Observation Programme. ESA has also prepared a proposal for an expanded data relay service, called GlobeNet (nothing to do with the same named submarine telecommunications cable system linking the USA, Bermuda, Venezuela and Brazil), that would provide support to the adoption by customers beyond Copernicus (previously GMES) and has proposed a third node in GEO orbit to provide redundancy and increase the availability and the flow of data from Earth observation satellites and UAVs to users, as well as to enhance near real-time services, including GEO-GEO O-ISL (optical inter-satellite link) capabilities.
So it looks like in a few years time, ESA's own and commercially available space-to-space capable tracking and data relay system, called either like it is now (EDRS - European Data Relay System), or by the already taken name of GlobeNet, will join TDRSS.