Unfortunately the other answer is inaccurate (only for Cisco switches or other switches doing per VLAN spanning tree and using the VLAN as extended system ID): yes, it can very well make a difference. Granted, under rather specific circumstances.

As you said, the bridge-id is made up basically of three parts: (priority + system id extension).mac-address, where the MAC address is just appended. The lowest bridge IDs wins the root bridge election. That being said, it is obvious that the priority must be the same on all routers in order for the extended system ID to actually make a difference.

Suppose a network just consisting of two switches. SW1 has a MAC address of 1 (just going to short that up here, suppose those were actually 48bit MAC addresses), SW2 has the MAC address 2. Both use the default priority (32768).

When running only PVSTP (no matter if rapid or not), the VLAN ID of that instance is used as extended system ID. In that case (for the sake of that example we're running in VLAN 1), the bridge IDs would be

SW1: 32769.1   (32768 + 1).MAC
SW2: 32769.2   (32768 + 1).MAC

Obviously SW1 will become root. Suppose now, SW2 was changed to MST mode and SW1 is still running RSTP. MSTP is fully backwards compatible to other spanning tree protocols. With MSTP, the extended system is becomes the MST instance number, as it would make no sense to use the VLAN here. Many VLANs can be mapped to a single instance. Just enabling MSTP without any configuration will create the MST instance 0 (!) and map all VLANs to it. Now the bridge IDs would look as follows:

SW1: 32769.1   (32768 + 1).MAC
SW2: 32768.2   (32768 + 0).MAC

Now, SW2 will be the winner of the election process, and it wins due to the lower extended system id. Hence the answer to your question is: yes, it could possibly affect root bridge election. The lowest possible VLAN number is 1, and the default MST instance is 0, resulting in the MST switch always winning root election with default settings.

If you think about that for a minute, that makes perfect sense. MST + RPVSTP interoperability requires the root bridge to be within the MST domain. Sure you can manually tweak the priority, in which case you messed that up yourself. By default however, the switch in the MST domain will be selected as root bridge.

TL;DR and Summary:

If all switches are running (R)PVSTP or all switches are running MSTP, then no, it will not affect root bridge election. In a mixed environment, it certainly can.

One note, though: this discussion is certainly true for Cisco switches (more general, switches supporting per VLAN spanning tree), but not necessarily for other switches.

Benjamin Dales answer is certainly correct for switches running standards based spanning tree. For switches using per VLAN STP however: yes, it can very well make a difference. Granted, under rather specific circumstances.

As you said, the bridge-id is basically made up of three parts: (priority + system id extension).mac-address, where the MAC address is just appended. The lowest bridge IDs wins the root bridge election. That being said, it is obvious that the priority must be the same on all routers in order for the extended system ID to actually make a difference.

Suppose a network just consisting of two switches. SW1 has a MAC address of 1 (just going to short that up here, suppose those were actually 48bit MAC addresses), SW2 has the MAC address 2. Both use the default priority (32768).

When running only PVSTP (no matter if rapid or not), the VLAN ID of that instance is used as extended system ID. In that case (for the sake of that example we're running in VLAN 1), the bridge IDs would be

SW1: 32769.1   (32768 + 1).MAC
SW2: 32769.2   (32768 + 1).MAC

Obviously SW1 will become root. Suppose now, SW2 was changed to MST mode and SW1 is still running RSTP. MSTP is fully backwards compatible to other spanning tree protocols. With MSTP, the extended system is becomes the MST instance number, as it would make no sense to use the VLAN here. Many VLANs can be mapped to a single instance. Just enabling MSTP without any configuration will create the MST instance 0 (!) and map all VLANs to it. Now the bridge IDs would look as follows:

SW1: 32769.1   (32768 + 1).MAC
SW2: 32768.2   (32768 + 0).MAC

Now, SW2 will be the winner of the election process, and it wins due to the lower extended system id. Hence the answer to your question is: yes, it could possibly affect root bridge election. The lowest possible VLAN number is 1, and the default MST instance is 0, resulting in the MST switch always winning root election with default settings. Also please have a look at Benjamins comment about spanning interoperability. I am currently not sure about the behavior (on BPDU sending level) when MSTP and PVSTP+ have to work together. I will add detailed information later, I have to try that in lab and inspect the BPDUs.

If you think about that for a minute, that makes perfect sense. MST + RPVSTP interoperability requires the root bridge to be within the MST domain. Sure you can manually tweak the priority, in which case you messed that up yourself. By default however, the switch in the MST domain will be selected as root bridge.

TL;DR and Summary:

If all switches are running (R)PVSTP or all switches are running MSTP, then no, it will not affect root bridge election. In a mixed environment, it certainly can.

One note, though: this discussion is certainly true for Cisco switches (more general, switches supporting per VLAN spanning tree), but not necessarily for other switches.